The Peninei Halachah --which enjoys widespread popularity in Israel & are used in schools and military academies alike-- justifies this "Mitzvah to wipe out Amalek" by referencing Samuel 15:1-3 where God commands the Israelites to "totally destroy all that belongs to [Amalek]" including "men and women, children and infants, cattle and sheep, camels and donkeys."
https://en.m.wikipedia.org/wiki/Eliezer_Melamed
https://ph.yhb.org.il/en/about-2/
https://www.biblegateway.com/passage/?search=1%20Samuel%2015%3A1-3&version=NIV