"Palaistine" shows up first in Herodotus as a term for a part of Syria, apparently including the area where the Philistines had lived (exactly how much of 1947 Palestine was included in that term isn't clear). The name derives from the name of the Philistines but they had probably been totally assimilated into the Canaanite population by then (they seem to have originated in the Aegean area and are thought to be the same as the "Peleset" who were one of the Sea Peoples who attacked Egypt around 1200 BC, and then settled in the region where they are found in the Bible). When Herodotus was writing that was part of the Persian Empire but perhaps the Persian name derived from terms used by earlier empires (Assyrians, Babylonians).
Gat is better known as Gath in English translations of the Bible. In I Samuel chapter 5 the Ark of the Covenant is briefly there, and the men are punished with hemorrhoids.
David took refuge in Gath (I Samuel 27) when hiding from King Saul.
After Saul's death, when David is lamenting Saul and Jonathan, he says, "Tell it not in Gath, publish it not in the streets of Askelon, lest the daughters of the Philistines rejoice..." (II Samuel 1.20).