Is the Society of St. Pius X schismatic or not?

The SSPX was founded as a legitimate organization within the Church during Pope Paul VI's reign.

Faithful can attend SSPX Masses without committing a schismatic act nor incurring excommunication. See, for example, the case of "The Hawaii Six," where then-Cardinal Ratzinger nullified the Hawaiian bishop's unjust "excommunications" of six SSPX faithful.

In 2009, Pope Benedict XVI lifted the excommunications of the SSPX bishops.

On 1 Sep. 2015, Pope Francis gave the SSPX priests jurisdiction to hear confessions and absolve sins during the Year of Mercy:

A final consideration concerns those faithful who for various reasons choose to attend churches officiated by priests of the Fraternity of St Pius X [SSPX]. This Jubilee Year of Mercy excludes no one. From various quarters, several Brother Bishops have told me of their good faith and sacramental practice, combined however with an uneasy situation from the pastoral standpoint. I trust that in the near future solutions may be found to recover full communion with the priests and superiors of the Fraternity. In the meantime, motivated by the need to respond to the good of these faithful, through my own disposition, I establish that those who during the Holy Year of Mercy approach these priests of the Fraternity of St Pius X to celebrate the Sacrament of Reconciliation shall validly and licitly receive the absolution of their sins.

Thus, the SSPX is not schismatic.

Societies of Apostolic Life, "without taking religious vows, pursue the apostolic purpose proper to each society. Living a fraternal life in common in their own special manner, they strive for the perfection of charity through the observance of the constitutions" [Canon 731]. The Society of S Pius X is constituted as a Society of Apostolic Life with the canonical approval of the Bishop of Fribourg [Canon 579]. Such Societies, "since they are dedicated in a special way to the service of God and of the whole Church, are in a particular manner subject to its supreme authority" [Canon 590].

The Society itself says

The Society of St. Pius X humbly submits to the authority of the pope, supreme guardian of the faith, and pays him all the reverence due to the chosen head of God’s Church.

If and when the time-honored teachings of the Church are obscured or perhaps even seemingly contradicted by the authorities in Rome, however, the SSPX stands firm for tradition.

Archbishop Lefebvre consecrated four priests as bishops in 1998 without authority, an act which was schismatic under Canon Law [Canon 751] and resulted in all five being excommunicated by Pope John Paul II.

In 2009, Pope Benedict XVI remitted the excommunication of the four bishops consecrated without authority by Archbishop Lefebvre, and explained in an apostolic letter:

An episcopal ordination lacking a pontifical mandate raises the danger of a schism, since it jeopardizes the unity of the College of Bishops with the Pope. Consequently the Church must react by employing her most severe punishment – excommunication – with the aim of calling those thus punished to repent and to return to unity. Twenty years after the ordinations, this goal has sadly not yet been attained. The remission of the excommunication has the same aim as that of the punishment: namely, to invite the four Bishops once more to return.

He went on

This gesture was possible once the interested parties had expressed their recognition in principle of the Pope and his authority as Pastor, albeit with some reservations in the area of obedience to his doctrinal authority and to the authority of the Council.

This mirrors the Society's own statement.

Pope Benedict then dealt with the status of the Society and its ministers:

As long as the Society does not have a canonical status in the Church, its ministers do not exercise legitimate ministries in the Church. There needs to be a distinction, then, between the disciplinary level, which deals with individuals as such, and the doctrinal level, at which ministry and institution are involved. In order to make this clear once again: until the doctrinal questions are clarified, the Society has no canonical status in the Church, and its ministers – even though they have been freed of the ecclesiastical penalty – do not legitimately exercise any ministry in the Church.

Thus although the ministers within the SSPX who were illegitimately consecrated have been released from the individual penalty of that schismatic act, the fact that they belong to an organisation constituted as a Society of Apostolic Life which has reservations about the authority of the Pope and the Second Vatican Council [cf Canon 590] means that they cannot minister within the Church: they remain under interdict.

Because the Society itself (in the second sentence quoted at the top of this answer) has set itself to be the authority to determine if "time-honoured teachings are obscured or seemingly contradicted," it remains schismatic.