To hit with a bat or (figuratively) as if with a bat.
He batted the ball away with a satisfying thwack.
To take a turn at hitting a ball with a bat in sports like cricket, baseball and softball, as opposed to fielding.
To strike or swipe as though with a bat.
The cat batted at the toy.
To flutter
bat one's eyelashes
To wink.
To bate or flutter, as a hawk.
(usually with 'around' or 'about') To flit quickly from place to place.
I've spent all week batting around the country.