ban (v.)
Old English bannan "to summon, command, proclaim," from Proto-Germanic *bannan "proclaim, command, forbid" (cognates: Old High German bannan "to command or forbid under threat of punishment," German bannen "banish, expel, curse"), originally "to speak publicly," from PIE root *bha- (2) "to speak" (cognates: Old Irish bann "law," Armenian ban "word;" see fame (n.)).

Main modern sense of "to prohibit" (late 14c.) is from Old Norse cognate banna "to curse, prohibit," and probably in part from Old French ban, which meant "outlawry, banishment," among other things (see banal) and was a borrowing from Germanic. The sense evolution in Germanic was from "speak" to "proclaim a threat" to (in Norse, German, etc.) "curse."

The Germanic root, borrowed in Latin and French, has been productive: banish, bandit, contraband, etc. Related: Banned; banning. Banned in Boston dates from 1920s, in allusion to the excessive zeal and power of that city's Watch and Ward Society.
ban (n.2)
"governor of Croatia," from Serbo-Croatian ban "lord, master, ruler," from Persian ban "prince, lord, chief, governor," related to Sanskrit pati "guards, protects." Hence banat "district governed by a ban," with Latinate suffix -atus. The Persian word got into Slavic perhaps via the Avars.
ban (n.1)
"edict of prohibition," c.1300, "proclamation or edict of an overlord," from Old English (ge)bann "proclamation, summons, command" and Old French ban, both from Germanic; see ban (v.).