Was there an export industry of papyrus from ancient Egypt?

Papyrus was known to the Greek world since the 8^th century BC, as it's mentioned in the Odyssey:

[Hom. Od. 21.390] Now there lay beneath the portico the cable of a curved ship, made of byblus plant, wherewith he made fast the gates, and then himself went within. Thereafter he came and sat down on the seat from which he had risen, and gazed upon Odysseus; now he was already handling the bow, turning it round and round, and trying it this way and that, lest worms might have eaten the horns, while its lord was afar. And thus would one speak with a glance at his neighbor: “Verily he has a shrewd eye, and is a cunning knave with a bow. It may be haply that he has himself such bows stored away at home, or else he is minded to make one, that he thus turns it this way and that in his hands, the rascally vagabond.”

The scene described takes place in Ithaca after Odysseus return, and "the cable of a curved ship, made of byblus plant" suggests that the Greeks were at the time familiar with the various uses of cyperus papyrus, the papyrus plant.

Moving on, Herodotus starts his Histories by describing an event that involved Phoenician traders carrying Egyptian and Assyrian merchandise in Greece:

[Hdt. 1.1.1] The Persian learned men say that the Phoenicians were the cause of the dispute. These (they say) came to our seas from the sea which is called Red, and having settled in the country which they still occupy, at once began to make long voyages. Among other places to which they carried Egyptian and Assyrian merchandise, they came to Argos,

[Hdt. 1.1.2] which was at that time preeminent in every way among the people of what is now called Hellas. The Phoenicians came to Argos, and set out their cargo.

[Hdt. 1.1.3] On the fifth or sixth day after their arrival, when their wares were almost all sold, many women came to the shore and among them especially the daughter of the king, whose name was Io (according to Persians and Greeks alike), the daughter of Inachus.

[Hdt. 1.1.4] As these stood about the stern of the ship bargaining for the wares they liked, the Phoenicians incited one another to set upon them. Most of the women escaped: Io and others were seized and thrown into the ship, which then sailed away for Egypt.

Trading between the two civilizations became more regular in 570 BC, when Naukratis was established in the Nile delta. Naukratis was the only permanent Greek colony in Egypt until the Ptolemaic times, and Egypt's larger harbour. Papyrus was one of the commodities regularly traded through Naukratis and Herodotus, who probably spend some time in the city during his visit in Egypt, gives us a good description of how the Egyptians... cooked it:

[Hdt. 2.92.5] They also use the byblus which grows annually: it is gathered from the marshes, the top of it cut off and put to other uses, and the lower part, about twenty inches long, eaten or sold. Those who wish to use the byblus at its very best, roast it before eating in a red-hot oven. Some live on fish alone. They catch the fish, take out the intestines, then dry them in the sun and eat them dried.

The word "byblus" (βύβλος/βίβλος) was the common Greek name of the papyrus plant and its use supports Herodotus story of Phoenician traders introducing the plant and other Egyptian commodities to the Greeks as it comes from the Greek name of the Phoenician city Gebal, the oldest continuously inhabited city in the world. Theophrastus in Enquiry into Plants used the word "papuros" when referring to the plant used as foodstuff.

In all probability, papyrus was imported from Egypt and not grown in Greece, all authors that mention the plant, including Theophrastus and Pliny the Elder, note that it was grown in Egypt. Nevertheless, Demosthenes, in Against Dionysodorus, mentions that paper (i.e. papyrus) in 4^th century BC Athens was relatively cheap and in common use:

[Dem. 56 1] I am a sharer in this loan, men of the jury. We, who have engaged in the business of overseas trade and put our money in the hands of others, have come to know one thing very clearly: that in all respects the borrower has the best of us. He received the money in cash which was duly acknowledged, and has left us on a scrap of paper which he bought for a couple of coppers, his agreement to do the right thing. We on our part do not promise to give the money, we give it outright to the borrower.

Later on we learn that Dionysodorus' trade route was from Athens to Egypt via Rhodes:

[Dem. 56 3] Now Dionysodorus here does neither the one nor the other, but has come to such a pitch of audacity, that although he borrowed from us three thousand drachmae upon his ship on the condition that it should sail back to Athens, and although we ought to have got back our money in the harvest-season of last year, he took his ship to Rhodes and there unladed his cargo and sold it in defiance of the contract and of your laws; and from Rhodes again he despatched his ship to Egypt, and from thence back to Rhodes, and to us who lent our money at Athens he has up to this day neither paid back our money nor produced to us our security.

[Dem. 56 5] This Dionysodorus, men of Athens, and his partner Parmeniscus came to us last year in the month Metageitnion, and said that they desired to borrow money on their ship on the terms that she should sail to Egypt and from Egypt to Rhodes or Athens, and they agreed to pay the interest for the voyage to either one of these ports.

The speech gives us a pretty good idea of how established the trade route was, with several financial and logistical details, suggesting that, at the time, it was a common route. Dionysodorus cargo was grain, but it's not hard to imagine that papyrus would reach Athens and Greece in general from a similar, or even the same, route.