paddle (n.)
c.1400, padell "small spade," from Medieval Latin padela, of uncertain origin, perhaps from Latin patella "small pan, little dish, plate," diminutive of patina (see pan (n.)).

Meaning "short oar with a wide blade" is from 1620s. As an instrument used for beating clothes (and slaves, and schoolboys), it is recorded from 1828, American English. Paddle-ball attested from 1935.
paddle (v.1)
"to dabble, wade in water," 1520s, probably cognate with Low German paddeln "tramp about," frequentative of padjen "to tramp, to run in short steps," from pad (v.). Related: Paddled; paddling. Meaning "to move in water by means of paddles" is a different word (see paddle (v.3)).
paddle (v.2)
"to beat with a paddle, spank," 1856, from paddle (n.). Related: Paddled; paddling.
paddle (v.3)
"to move in water by means of paddles," 1670s, from paddle (n.). To paddle one's (own) canoe "do for oneself" is from 1828.