Yes, fish sleep.  Because fish have no eyelids, they cannot close their eyes—and it may look as though they never sleep.  Many fish sleep by simply remaining very still in the water.

Others lie on the bottom, resting upright or on their sides.  And a few others wriggle into the sand on the floor of the body of water in which they live.

In this way, fish rest their bodies as you and I do when we sleep.  If you happen to see a fish lying almost motionless in the water, it is probably asleep.  Fish don’t sleep as soundly as humans do, and the slightest disturbance will wake them.–Dick Rogers