In 2012 Daniel Wallace mentioned in a debate the existence of an early fragment of the Gospel of Mark. LiveScience has an update with information from Craig Evans:

A text that may be the oldest copy of a gospel known to exist — a fragment of the Gospel of Mark that was written during the first century, before the year 90 — is set to be published.
[…]
This first-century gospel fragment was written on a sheet of papyrus that was later reused to create a mask that was worn by a mummy.
[…]
Evans says that the text was dated through a combination of carbon-14 dating, studying the handwriting on the fragment and studying the other documents found along with the gospel. These considerations led the researchers to conclude that the fragment was written before the year 90. With the nondisclosure agreement in place, Evans said that he can’t say much more about the text’s date until the papyrus is published.
[…]
Evans said that the research team will publish the first volume of texts obtained through the mummy masks and cartonnage later this year. It will include the gospel fragment that the researchers believe dates back to the first century.

The full story is here.

HT: Craig Dunning

UPDATE: There are some errors in this LiveScience article. See the post by Peter Williams (together with the comments) here. (HT: Ulrich Wendel)