sleeve (n.)
Old English sliefe (West Saxon), slefe (Mercian) "arm-covering part of a garment," probably literally "that into which the arm slips," from Proto-Germanic *slaubjon (cognates: Middle Low German sloven "to dress carelessly," Old High German sloufen "to put on or off"). Related to Old English slefan, sliefan "to slip on (clothes)" and slupan "to slip, glide," from PIE root *sleubh- "to slide, slip."

Compare slipper, Old English slefescoh "slipper," slip (n.) "woman's garment," and expression to slip into "to dress in"). Mechanical sense is attested from 1864. To have something up one's sleeve is recorded from c.1500 (large sleeves formerly doubled as pockets). Meaning "the English Channel" translates French La Manche.