Archer, that is the usual response that I get from folks who believe in the Bible, and it is certainly logically possible that the Hebrew myth was older. Here's the problem, though. The Gilgamesh epic existed even before there were records of Semitic speakers. The first Semites in recorded history, were the Akkadians, who adopted and embellished the Gilgamesh story. (It was "Bilgamesh" in Sumerian.) The impact of the Akkadians on the region was huge, and the Gilgamesh epic was widely known. Hebrew speakers appear as a minor set of tribes much later in history. Their version was spread primarily through an oral tradition, so we do not know all of the editorial changes that it went through before it was written down and codified in Greek (and later in Aramaic and Hebrew). The Akkadian and Sumerian records, written on tablets and carved in stone, come down to us completely unedited.
When the Ugaritic language was finally discovered and deciphered in the last century, it also contained tales of Gilgamesh. What surprised many scholars was that large parts of the Hebrew Psalms and other portions of the Hebrew Bible existed almost verbatim in the pagan literature of the Ugarits. That suggests that people living in Palestine to the south of Ugarit shared the same folklore and traditions as the Ugarits. It's just that the Hebrew story appears to have been customized to fit their particular history and position in the local politics of the time. Ultimately, Hebrew speakers came to be monotheistic, but that probably was not even the case when the earliest versions of biblical stories came to be recorded in the histories of Israel and Judea. It was fairly common back then for states and empires to make up aggrandized tales of their origins, and the Hebrew empire under David and Solomon was probably little different in that regard.