Did the early church believe that St. Peter was the first pope?

As usual, it is important to define terms like the papacy. Here is New Advent.

This term is employed in an ecclesiastical and in an historical signification. In the former of these uses it denotes the ecclesiastical system in which the pope as successor of St. Peter and Vicar of Jesus Christ governs the Catholic Church as its supreme head. -NewAdvent-

From a Catholic viewpoint, the primacy of the bishop of Rome is largely derived from his role as the apostolic successor to Saint Peter, to whom primacy was conferred by Jesus, who gave Peter the Keys of Heaven and the powers of "binding and loosing", naming him as the "rock" upon which the Church would be built. -source-

So, papacy is the idea that the Church at Rome governs the whole church as its head. Did the very early church believe this?

If they believed it, what might we find? Proclamation from Rome and agreement among the remainders. We don't find this.

16. And when the blessed Polycarp was at Rome1713 in the time of Anicetus, and they disagreed a little about certain other things, they immediately made peace with one another, not caring to quarrel over this matter. -Eusebius Book 5 Chapter 24

1. But the bishops of Asia, led by Polycrates, decided to hold to the old custom handed down to them.1695 He himself, in a letter which he addressed to Victor and the church of Rome, set forth in the following words the tradition which had come down to him:1696 -ibid-

So, on this issue circa 150-200 CE, it is clear that bishops rejected Rome's claim to supremacy.

In 256 CE, Cyprian and Firmillian rejected Rome's claim that baptism apart from the church was efficacious.

6. But that they who are at Rome do not observe those things in all cases which are handed down from the beginning, and vainly pretend the authority of the apostles;2925 any one may know also from the fact, that concerning the celebration of Easter, and concerning many other sacraments of divine matters, he may see that there are some diversities among them, and that all things are not observed among them alike, which are observed at Jerusalem, just as in very many other provinces also many things are varied because of the difference of the places and names.2926 And yet on this account there is no departure at all from the peace and unity of the Catholic Church, such as Stephen has now dared to make;2927 breaking the peace against you, which his predecessors have always kept with you in mutual love and honour, even herein defaming Peter and Paul the blessed apostles,2928 as if the very men delivered this who in their epistles execrated heretics, and warned us to avoid them. Whence it appears that this tradition is of men which maintains heretics, and asserts that they have baptism, which belongs to the Church alone. -Letter from Firmillian-

Strong language.

Irenaeus is typically cited as someone who suggests all should agree with whatever comes out from Rome.

1. It is within the power of all, therefore, in every Church, who may wish to see the truth, to contemplate clearly the tradition of the apostles manifested throughout the whole world; and we are in a position to reckon up those who were by the apostles instituted bishops in the Churches, and [to demonstrate] the succession of these men to our own times; those who neither taught nor knew of anything like what these [heretics] rave about. For if the apostles had known hidden mysteries, which they were in the habit of imparting to “the perfect” apart and privily from the rest, they would have delivered them especially to those to whom they were also committing the Churches themselves. For they were desirous that these men should be very perfect and blameless in all things, whom also they were leaving behind as their successors, delivering up their own place of government to these men; which men, if they discharged their functions honestly, would be a great boon [to the Church], but if they should fall away, the direst calamity. -Irenaeus-

Note the bolded parts. Irenaeus speaks plainly that the apostolic tradition is widespread, found throughout the world. He goes on to speak about Rome, but also about Ephesus and Smyrna. His point is that there is no supreme head, besides Christ, but continues the idea of Acts 15 of a council of churches. Inasmuch as the apostolic tradition is handed down from faithful men to faithful men. Plural.

So, to answer the OP. No, the early church did not submit to the idea of a papacy, a supreme earthly head.

Points for consideration.

(1.) The vast majority of modern Roman Catholic historians acknowledge that there was no monarchical episcopate (a single ruling bishop rather than a plurality of elders) in the city of Rome until at least the middle of the second century. Eamon Duffy, a Roman Catholic Historian and former member of the Pontifical Historical Commission, is representative of this consensus:

To begin with, indeed, there was no ‘pope’, no bishop as such, for the church in Rome was slow to develop the office of chief presbyter, or bishop. By the end of the first century the loose pattern of Christian authority of the first generation of believers was giving way in many places to the more organised rule of a single bishop for each city, supported by a college of elders. …There is no sure way to settle on a date by which the office of ruling bishop had emerged in Rome, and so to name the first Pope, but the process was certainly complete by the time of Anicetus in the mid-150s…¹

For extensive primary source documentation of this consensus (from Roman Catholic Scholars) see Appendix A.

(2.) While the petrine office would develop slowly over time in the west it was never accepted by the eastern Churches. Ronald Minnerath, Roman Catholic Historian and Archbishop of Dijon, writes:

In the first millennium there was no question of the Roman bishops governing the church in distant solitude. They used to take their decisions together with their synod, held once or twice a year. When matters of universal concern arose, they resorted to the ecumenical council. Even Leo, who struggled for the apostolic principle over the political one, acknowledged that only the emperor would have the power to convoke an ecumenical council and protect the church. At the heart of the estrangement that progressively arose between East and West, there may be a historical misunderstanding. The East never shared the Petrine theology as elaborated in the West. It never accepted that the prōtos in the universal church could claim to be the unique successor or vicar of Peter. So the East assumed that the synodal constitution of the church would be jeopardized by the very existence of a Petrine office with potentially universal competencies in the government of the church.²

(3.) The Patristic writers, outside of the Church at Rome, never understood Matthew 16:18 to be speaking of petrine succession or authority. Yves Congar, O.P., Roman Catholic historian and the Cardinal Deacon of the Basilica of San Sebastiano al Palatino, writes:

…the interpretation of Peter’s confession in Matthew 16.16-19. Except at Rome, this passage was not applied by the Fathers to the papal primacy; they worked out exegesis at the level of their own ecclesiological thought, more anthropological and spiritual than judicial. ³

John Chrysostom (c. 347-407), the Archbishop of Constantinople, is representative of the Patristic consensus regarding Matthew 16:18:

He did not say upon Peter for it is not upon the man, but upon his own faith that the church is built. And what is this faith? You are the Christ, the Son of the living God.⁴

For extensive primary source documentation of this consensus see: William Webster, The Matthew 16 Controversy: Peter and the Rock, (Battle Ground: Christian Resources Inc., 1996; second printing, revised 1999). Most of which may be read online Here and Here.

Appendix A:

Francis A Sullivan, S.J.:

Scholars differ on details, such as how soon the church of Rome was led by a single bishop, but hardly any doubt that the church of Rome was still led by a group of presbyters for at least a part of the second century.⁵

William J. La Due, J.C.D.:

…it is now quite generally accepted that the monarchical episcopate in Rome did not originate much before 140-150 A.D.⁶

Robert B. Eno, S.S.:

A new and important study by Peter Lampe draws the picture of the early Roman community divided into a number of smaller house churches scattered throughout the city and its environs, each presided over by a presbyter or perhaps more than one). There was really no united and coordinated Church leadership ad intra, i.e., within the city’s Christian community as a whole. …This evidence (Clement, Hermas, Ignatius) points us in the direction of assuming that in the first century and into the second, there was no bishop of Rome in the usual sense given to that title. The office of the single mon-episkopos was slowly emerging in the local Christian communities around the Mediterranean world. Men like Ignatius were strongly urging this development. But the evidence seems to indicate that in the earliest decades, this evolution had not yet been accomplished in Rome. This then is that missing link referred to by Rudolf Pesch. If there were no bishop of Rome, in what sense can one speak of a Petrine succession?⁷

J. Michael Miller, C.S.B. (Archbishop of Vancouver):

While admitting that the monarchical episcopate came about as the result of historical choices, Catholic doctrine holds that its emergence was guided by the Spirit. …the monarchical episcopacy was not a universal and normative ecclesial structure before the mid-second century.⁸

John P. Meier, S.S.D.:

…it is not without significance that neither the Catechism of the Catholic Church nor Pope John Paul II’s groundbreaking encyclical Ut unum sint employed certain problematic assertions like “St. Peter was the first Pope.” Granted, academics may smile at such an assertion, yet it is still often heard in the popular media, to say nothing of homilies and catechetical instruction. Hence it is at least noteworthy that some recent authoritative documents of the Roman Catholic Church have avoided certain types of claims that would not hold up under the scrutiny of critical historical research. …In striking contrast to 1 and 2 Peter, where individual teachers speak not only in the name of Peter but as Peter, the anonymous author of 1 Clement speaks as “we,” as the whole church of Rome, which presumably, like Corinth, is led by a group of presbyter-bishops assisted by deacons. There is no sign of the monepiscopate soon to be championed by Ignatius. It is rather “the church of God sojourning in Rome” as such, and not Peter, and not some individual claiming to be the successor of Peter, that implicitly exercises authority over a distinguished Pauline church in Greece. Indeed, the final blessing of the epistle is extended not only to the Corinthians but also “to all those called by God everywhere” (1 Clement 65.2).⁹

Klaus Schatz, S.J.:

In fact, this “letter of Clement,” written around 95, is the first document indicating that the Roman community felt responsible for other churches. Its name is a subsequent addition, of course: according to Hegesippus's list of bishops Clement was bishop of Rome at that time, the third in succession. However, he is not named as the author of the letter; instead, the true sender is the Roman community. We probably cannot say for certain that there was a bishop of Rome at that time. It seems likely that the Roman church was governed by a group of presbyters from whom there very quickly emerged a presider or “first among equals” whose name was remembered and who was subsequently described as “bishop” after the mid-second century.¹⁰

Raymond E. Brown, S.S.:

One may support this conclusion from a convergence of scattered evidence: from the instructions that had to be given to Titus (1:5) in the Pastorals; from the failure to mention bishops in the Corinthian correspondence where it would have been logical to invoke their aid; from the failure to mention presbyters in any undisputed Pauline letter; from the need of Clement in I Corinthians 42-44 to strengthen the episcopate/presbyterate by giving it a pedigree; from the evidence of Didache 15 that only gradually did bishops and deacons take over the functions of prophets and teachers (mentioned in I Cor 12:28; Eph 4:11).¹¹

Hermann J. Pottmeyer:

Anyone who wishes to come to an understanding of the papal ministry cannot avoid dealing with the history of this ministry. The historical facts are not disputed, but their theological evaluation is contentious. …It is sociologically explicable that leadership functions became necessary in the congregations, whether in regard to doctrine or discipline, or to representation to the outside world. In order that these responsibilities could be exercised permanently and in an orderly manner, the functions became offices. Among the various leadership models, that of the monepiscopate prevailed as the most effective. That evolved into the monarchical episcopate as the bishop combined in one individual the functions of the teacher of the congregation, of its leader or pastor, and of its priest, presiding at the celebration of the Eucharist.¹²

Ronald Minnerath (Archbishop of Dijon):

With these second-century witnesses, attention is drawn not on the person of Peter or on a successor in a time when monarchical episcopacy was only beginning to emerge. The general consideration is that the Church of Rome is the apostolic church par excellence, and the reference for the apostolic teaching. Indeed during the second century, it was the apostolic origin of a church that conferred on it a particular authority in the transmission of the “rule of faith.”¹³

Allen Brent:

Where we do have concrete information, as in the case of Clement in the third place after St Peter according to Irenaeus, the notion of the office of a single bishop in succession to a predecessor seems lacking. In his genuine letter to the Corinthians (c. AD 95), Clement does not write in his own name but in the name of the ‘church of God whose pilgrim residence is at Rome to the church of God residing similarly at Corinth’. There is no ‘Clement bishop, servant of the servants of God’ claiming apostolic authority for his office as successor to St Peter. He is writing, as has been pointed out, not as a single monarch-bishop but as the secretary of the Roman presbyterate.¹⁴

George Edward Dolan, S.T.L.:

It would appear that St. Jerome in the fourth century unwittingly laid the foundation when he wrote a defense of the presbyterate against the arrogance and abuses of certain Roman deacons. In order to restore to the presbyterate its rightful place and authority Jerome pointed out that in the very early days of the Church the terms episcopus and presbyter signified the same individuals. In other words, as we interpret Jerome all were bishops in the sense in which this word is understood today, with full powers to confirm and ordain. But when the universal monarchical episcopate was introduced into the government of the Church only the chief priest (ie., the bishop) was given the full powers of confirming and ordaining, while all other priests who were subjected to him (in other words, the presbyters) were given only a limited or restricted share in the powers of the priesthood.¹⁵

Notes:

^{1. Eamon Duffy, Saints & Sinners: A History of the Popes, (New Haven and London: Yale University Press, 2006), pp. 9-10, 13.}

^{2. James Puglisi, S.A., ed., How Can the Petrine Ministry Be a Service to the Unity of the Universal Church? (Grand Rapids: William B. Eerdmans, 2010), Archbishop Ronald Minnerath, “The Petrine Ministry in the Early Patristic Tradition,” pp. 47-48.}

^{3. Yves Congar, O.P., Tradition and Traditions: An Historical and a Theological Essay, (New York: The Macmillan Company, 1967), pp. 398-399. Cf. George Salmon: "The most elaborate examination of the opinions of the Fathers is in an Epistle [fn. *: Epist. vii., Opp. vol. v., pt. 2. p. 99: Geneva, 1731.] by the French Roman Catholic Launoy, in which, besides the interpretation that Peter was the rock, for which he produces seventeen Patristic testimonies, he gives the interpretations that the rock was the faith which Peter confessed, supported by forty-four quotations; that the rock was Christ Himself, supported by sixteen; and that the Church was built on all the Apostles, supported by eight." {George Salmon, The Infallibility of the Church, (London: John Murray, 1888), p. 329f.}}

^{4. J. P. Minge, Patrologiæ Cursus Completus: Patrologiæ Græcæ: Tomus LII, (Parisiis: 1862), Joannis Chrysostomi, Ad Homilias In Pentecosten, Sermo I, col. 806, 807; trans. Laurent Cleenewerck, His Broken Body, (Euclid University Press, 2008), p. 263. Cf. Augustine, Bishop of Hippo (c. 354-430 A.D.): "Christ, you see, built his Church not on a man but on Peter’s confession. What is Peter’s confession? You are the Christ, the Son o f the living God. There’s the rock for you, there’s the foundation, there’s where the Church has been built, which the gates of the underworld cannot conquer." {The works of Saint Augustine, A Translation for the 21st Century: Sermons: III/6 (184-229Z), On the Liturgical Seasons, trans. Edmund Hill, O.P., ed. John E. Rotelle, O.S.A., (New Rochelle: New City Press, 1993), Sermon 229P, p. 327.}}

^{5. Francis A Sullivan, S.J., From Apostles to Bishops: The Development of the Episcopacy in the Early Church, (New York/Mahwah: The Newman Press, 2001), p. viii, cf. p. 217. Cf. The Shepherd of Hermas (c. 2nd century): "Therefore you will write two little books, and you will send one to Clement and one to Grapte. Then Clement will send it to the cities abroad, because that is his job. But Grapte will instruct the widows and orphans. But you yourself will read it to this city, along with the elders [πρεσβυτέρων] who preside over the church." {Michael W. Holmes, The Apostolic Fathers: Greek Texts and English Translations of Their Writings, (Grand Rapids: Baker Academic, 2007), The Shepherd of Hermas, Vision 2.4.8:3, p. 469.}}

^{6. William J. La Due, J.C.D., The Chair Of Saint Peter, (Maryknoll: Orbis Books, 1999), p. 26.}

^{7. Robert B. Eno, S.S., The Rise Of The Papacy, (Wilmington: Michael Glazier, 1990), pp. 28, 29.}

^{8. J. Michael Miller, C.S.B., The Shepherd And The Rock, (Huntington: Our Sunday Visitor, 1995), pp. 60, 61.}

^{9. James Puglisi, S.A., ed., How Can the Petrine Ministry Be a Service to the Unity of the Universal Church? (Grand Rapids: William B. Eerdmans, 2010), John P. Meier “Petrine Ministry in the New Testament and in the Early Patristic Traditions,” pp. 15, 25.}

^{10. Klaus Schatz, S.J., Papal Primary: From its Origins to the Present, trans. John A. Otto & Linda M. Maloney, (Collegeville: The Liturgical Press, 1996), p. 4.}

^{11. Raymond Edward Brown, The Critical Meaning of the Bible, (New York: Paulist Press, 1981), n. 17, p. 134.}

^{12. James Puglisi, S.A., ed., How Can the Petrine Ministry Be a Service to the Unity of the Universal Church? (Grand Rapids: William B. Eerdmans, 2010), Hermann J. Pottmeyer “Historical Development of the Forms of Authority and Jurisdiction: The Papal Ministry — an Ecumenical Approach,” pp. 98, 99.}

^{13. James Puglisi, S.A., ed., How Can the Petrine Ministry Be a Service to the Unity of the Universal Church? (Grand Rapids: William B. Eerdmans, 2010), Archbishop Ronald Minnerath “The Petrine Ministry in the Early Patristic Tradition,” p. 36.}

^{14. Allen Brent, Ignatius of Antioch: A Martyr Bishop and the origin of Episcopacy, (London: Continuum, 2007), p. 125.}

^{15. George Edward Dolan, The Catholic University of America: Studies in Sacred Theology: (Second Series), No. 36: The Distinction Between the Episcopate and the Presbyterate According to the Thomistic Opinion: A Dissertation, (Washington, D.C.: The Catholic University of America Press, 1950), pp. 10-11. Cf. Jerome of Stridon (c. 342/7 - 420): "In writing both to Titus and to Timothy the apostle speaks of the ordination of bishops and of deacons, but says not a word of the ordination of presbyters; for the fact is that the word bishops includes presbyters also." {NPNF2, 6:289; The Letters of St. Jerome, Letter CXLVI, To Evangelus.}}

Καὶ αὐτός ἐστιν πρὸ πάντων καὶ τὰ πάντα ἐν αὐτῷ συνέστηκεν.

~ Soli Deo Gloria

It appears that the official use of the title of Pope (as given to Bishops of Rome) and confined to the successors of Peter, did not come into effect till the reign of Pope Gregory VII (1073-85).

The title of Pope was therefore applied posthumously from Peter on.

Peter nowhere claims supremacy over the other apostles. Nowhere in his writings (1 and 2 Peter) did the Apostle Peter claim any special role, authority, or power over the church. Nowhere in Scripture does Peter, or any other apostle, state that their apostolic authority would be passed on to successors. Yes, the Apostle Peter had a leadership role among the disciples. Yes, Peter played a crucial role in the early spread of the gospel (Acts chapters 1-10). Yes, Peter was the “rock” that Christ predicted he would be (Matthew 16:18). However, these truths about Peter in no way give support to the concept that Peter was the first pope, or that he was the “supreme leader” over the apostles, or that his authority would be passed on to the bishops of Rome. Peter himself points us all to the true Shepherd and Overseer of the church, the Lord Jesus Christ (1 Peter 2:25).

No Roman bishop claimed universal authority for himself until the mid-3rd century.

At the Council of Nicea, in A.D. 325, the bishop of Rome was given authority over an area similar to that of the bishop of Alexandria. The bishop of Alexandria had authority over all of Egypt.

From the early 3rd century the title was applied generically to all bishops. The earliest extant record of the word papa being used in reference to a Bishop of Rome dates to late 3rd century, when it was applied to Pope Marcellinus.

There does not appear to be any historical evidence from AD 100-200 AD that the early church had a pope, or a papacy, or that one man was the sole leader of the entire church.

This is a brief extract from an answer I gave to this question asked in February 2019:When was the formal title of Pope first officially bestowed on the Bishops of Rome?

This is the official Catholic source I used: https://www.newadvent.org/cathen/12260a.htm