The "George-Alow and the Sweepstake too is "A Dittye of the fight uppon the seas of the 4 of June last [1590 - ajs] in the Straytes of Jubraltare, betwene the George and the Thomas Bonaventure and eight Gailles with three ffregates"
From Oxford Book of Sea Songs, Palmer. This is Child #285,

Interesting song in that it completely dropped out of any recorded knowledge of tradition nor was any tune ever recorded until it suddenly reappeared at the beginning of this century as "High Barbary."

So you'll have to decide if this meets the definition.

As to Dibden & the many Frank Drake songs, I understand that very, very few of them actually went to sea. They were for & by landsmen. Most people define "sea songs" as those actually used at sea and mostly (but not necessarily) created by sailors. Dibden wrote "songs about the sea." Again, you define what you want.

KC, if you chance to read here, hi there. Now I'll never get Heise, All out of my head. Maybe playing the record a feew times will do it...

The Ballad Index gives 1765 (Percy) as earliest printing of "Patrick Spens." I haven't checked Roud.

There should be any number of period Spanish Armada (1588) songs, as well. Some in D'Urffey but if he wrote them, then they're too late, of course.

"A Ioyful New Ballad," (a broadside about the Armada) in A Ballad History of England, Roy Palmer may be period.

Then there's another question - What is the LAST provable chantey written?