There is a reference to Philistines in Genesis 21.32, in the time of Abraham, which is supposed to be before the Philistines settled in Palestine. In the Septuagint Greek version of the Bible, they are there called Phylistiim, but in many passages where the English translations have "Philistines" the Septuagint has "allophyloi" (foreigners).
Ezekiel 25.15-17 and Zechariah 9.5-7 predict destruction for the Philistines, which suggests there were still Philistines in the 6th century B.C. Gosta Ahlstrom, The History of Ancient Palestine, page 825, says the city of Ashkelon was Phoenician in the Persian period; Gaza was not Phoenician but "seems to have been part of the Arab economic sphere." Whether the people living in the old Philistine cities considered themselves Philistines is one thing; the Biblical authors may have continued to call them Philistines for historical reasons anyway.