I vaguely remember learning something in college about a written language being found in a cave not close to Sumer but dated to be older.
I'm not sure what current consensus is, but according to Samuel Kramer's The Sumerians: Their History, Culture and Character (written in the 50s) some words in Sumerian and their names of the rivers and cities come from the earlier, Ubaidian settlers who had a different language.
Source, page 40.
I do not think, though, that Ubaidians ever got to the point of inventing writing.