Are the Malay and Filipino words salamat and selamat derived from the Arabic assalam-alaikum?

GENERALLY, YES. The Arabic phrase "As-Salaam 3aleikum" (السلام عليكم) literally means "peace be with you" and is used as a formal greeting between Arabs and Non-Arab Muslims. The world selamat comes from the Arabic Word "selaama"/"selaamat" (سلامة), which means "safe" or "secure" and from the same root as "salaam" in "As-Salaam 3aleikum". This is why "selamat" means "safe" in Malay. However, because of the prevalence of "As-Salaam 3aleikum" among Muslims in Malaysia and the southern Philippines, "selamat" was also re-purposed in greetings as if it were "salaam" as well.