Thanks for your question.
Exodus 17:8-16 Then came Amalek, and fought with Israel in Rephidim. And Moses said unto Joshua, Choose us out men, and go out, fight with Amalek: to morrow I will stand on the top of the hill with the rod of God in mine hand. So Joshua did as Moses had said to him, and fought with Amalek: and Moses, Aaron, and Hur went up to the top of the hill. And it came to pass, when Moses held up his hand, that Israel prevailed: and when he let down his hand, Amalek prevailed. But Moses’ hands were heavy; and they took a stone, and put it under him, and he sat thereon; and Aaron and Hur stayed up his hands, the one on the one side, and the other on the other side; and his hands were steady until the going down of the sun. And Joshua discomfited Amalek and his people with the edge of the sword. And the LORD said unto Moses, Write this for a memorial in a book, and rehearse it in the ears of Joshua: for I will utterly put out the remembrance of Amalek from under heaven. And Moses built an altar, and called the name of it Jehovah Nissi: For he said, Because the LORD hath sworn that the LORD will have war with Amalek from generation to generation.
From the account above, the answer is NO. They were beaten in this battle but the Amalekites continued as a people. They were still in the land at the time of the Judges.
Judges 3:12,13 And the children of Israel did evil again in the sight of the LORD: and the LORD strengthened Eglon the king of Moab against Israel, because they had done evil in the sight of the LORD. And he gathered unto him the children of Ammon and Amalek, and went and smote Israel, and possessed the city of palm trees.
They were still in the land at the time of King Saul.
1 Samuel 15:2,7,8 Thus saith the LORD of hosts, I remember that which Amalek did to Israel, how he laid wait for him in the way, when he came up from Egypt. Now go and smite Amalek, and utterly destroy all that they have, and spare them not; but slay both man and woman, infant and suckling, ox and sheep, camel and ass.... And Saul smote the Amalekites from Havilah until thou comest to Shur, that is over against Egypt. And he took Agag the king of the Amalekites alive...
They were still in the land at the time of King David.
1 Chronicles 18:11 Them also king David dedicated unto the LORD, with the silver and the gold that he brought from all these nations; from Edom, and from Moab, and from the children of Ammon, and from the Philistines, and from Amalek. This is the last mention of Amalek apart from Psalm 83:7 Gebal, and Ammon, and Amalek; the Philistines with the inhabitants of Tyre;
Psalm 83 is a prophecy, so may be more concerned with the people that live in the land originally occupied by Amalek rather than the Amalekites themselves. As the king of the Amalekites was called Agag in the time of Saul and any Amalekites left would have gone into captivity at the time of Nebuchadnezzar, it may be that Haman the Agagite in Esther was in fact an Amalekite.
Esther 3:10 And the king took his ring from his hand, and gave it unto Haman the son of Hammedatha the Agagite, the Jews’ enemy.
I hope this helps,
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