1560s, "cotton padding," corrupted from earlier bombace (1550s), from Old French bombace "cotton, cotton wadding," from Late Latin bombacem, accusative of bombax "cotton, 'linteorum aut aliae quaevis quisquiliae,' " a corruption and transferred use of Latin bombyx "silk," from Greek bombyx "silk, silkworm" (which also came to mean "cotton" in Medieval Greek), from some oriental word, perhaps related to Iranian pambak (modern panba) or Armenian bambok, perhaps ultimately from a PIE root meaning "to twist, wind." From stuffing and padding for clothes or upholstery, meaning extended to "pompous, empty speech" (1580s).
Also from the same source are Swedish bomull, Danish bomuld "cotton," and, via Turkish forms, Modern Greek mpampaki, Rumanian bumbac, Serbo-Croatian pamuk. German baumwolle "cotton" is probably from the Latin word but altered by folk-etymology to look like "tree wool." Polish bawełna, Lithuanian bovelna are partial translations from German.