I can't speak outside of my own expertise (Latin and Greek), but I know of several manuscripts that outdate that by hundreds years!

The Ambrosian Iliad dates to the 5th Century CE, and we even have a sample of Cleopatra's signature from the 50s BCE!

The survivability of these artifacts depends greatly on the medium and the climate. Oxyrinchus, Egypt was a Grek-then-Roman dump that, because it was in the desert, meant that pieces of papyrus with writing survived nearly 1800 years intact. Recently, fragments of a lost poem of Sappho were discovered there. Admittedly, since archaeologists are essentially digging through ancient trash, these works are fragmentary.

Vellum, on the other hand, is much hardier but does not come into use until the Late Roman Empire and Middle Ages.

The oldest manuscripts still extant are papyrus fragments from dynastic Egypt. The very oldest date from about 2600 BC and were only discovered in 2013 in an ancient boat store, just west of the Red Sea. Amazingly, they could be dated to the very time the great pyramid of Giza was being constructed, during the reign of Khufu. One manuscript mentions limestone being transported up the Nile for the pyramid's construction at Giza, others mention provisions provided for the boat workers. Here is one of the papyri with writing in hieroglyphic script, others are written in hieratic, a less formal script.

In terms of a text still famous today we can look at the Bible, specifically the Old Testament books, the earliest manuscripts of which date to the 2nd century BC. Most are in fact written on treated leather, but they also include papyrus and copper scrolls. These are among the Dead Sea Scrolls, which were discovered in 1947 in caves west of the Dead Sea. The writing on these scrolls was quickly identified as ancient Hebrew, and analysis of the handwriting (paleography) indicated they dated from the early centuries BC to the 1st century AD. Radiocarbon dating and archeology further confirmed this and narrowed the texts down to about 200 BC to 70 AD. A nearby ancient settlement was identified as the site of the religious community who wrote the scrolls, a strict sect who had based themselves in the wilderness outside society, although they apparently retained links with related sectarians in Jerusalem. The site was abandoned during the Jewish wars against the Romans, presumably because the sect took part in the war, and a Roman fort was later build on the site. The community clearly used most of the books that make up the Old Testament today, the manuscripts and fragments represent every book of the Hebrew Bible except Esther.* Interestingly, they also included pseudo-Biblical works like the Book of Enoch, and many previously unknown works expanding on Biblical stories. The scrolls have proved very useful for reconstructing an earlier form of the Old Testament than was previously possible, and have illuminated a part of Jewish religion during the Maccabean and Roman periods.

---

* The Hebrew Bible (the Tanach) is equivalent to the Protestant Old Testament. Roman Catholic and Eastern Orthodox Old Testaments have more books, some of which are also represented in the Dead Sea Scrolls, e.g. the book of Tobit.

Sources

Alexander Stille (2015) The World’s Oldest Papyrus and What It Can Tell Us About the Great Pyramids. [online] Available at: https://www.smithsonianmag.com/history/ancient-egypt-shipping-mining-farming-economy-pyramids-180956619/

Geza Vermes (2011) The Complete Dead Sea Scrolls in English (7th edn). London: Penguin Books (especially pages 12-15)

Florentino G. Martinez (1996) The Dead Sea Scrolls Translated: The Qumran Texts in English (2nd edn). Leiden: E. J. Brill (especially pages xlv-xlviii)