‘Umar bin al-Khattāb, or ‘Umar I, was the second caliph after Muhammad. The vast majority of sources on his life are in Arabic and date to the ninth century or later. Before that point, we have little more than scraps.
The earliest source to mention ‘Umar I is an Arabic graffito from north-western Arabia (my translation):
I, Zuhayr, wrote this in the time when ‘Umar died, in the year twenty-four.
That is, 24 by the Muslim calendar; 644 by our reckoning. This ‘Umar was presumably someone well known, else he wouldn’t have been mentioned here; and later sources place ‘Umar’s death in the year 23, which corroborates the inscription very nicely. Beyond reasonable doubt, this inscription attests to the existence and importance of ‘Umar I.
‘Umar is also named in the seventh-century Armenian chronicle attributed to Sebeos (ch. 30, Bedrosian’s translation):
When the sons of Ishmael had arisen and issued from the desert of Sinai, their king Amrh did not accompany them. But when [the Arabs] had militarily routed both kingdoms … the [Arab] king gave an order to assemble boats and many sailors and to navigate southwardly.
Note that the manuscripts for this chronicle were produced in the seventeenth century or later. This is a common enough problem with medieval sources; in any case, there’s no obvious reason to suspect tampering with the passage above.
And his name appears in a short list, in Syriac, of “kings” of the Arabs, apparently composed in 705; albeit preserved in a manuscript from the late ninth century (Hoyland, Seeing Islam, 394-5):
…There reigned after [Muhammad] Abū Bakr for 2 years. And there reigned after him ‘Umūr for 12 years. And there reigned after him ‘Uthmān for 12 years, and they were without a leader … for 5 ½ years.
If you know of any other very early mentions of ‘Umar I, please send them my way.
Update: See also Rich Heffron’s compilation of early sources for Abū Bakr.