Yes and no. Dolphins sleep, but not in the way you and I define sleep.
Dolphins do not breathe spontaneously. In other words, they have to actively decide when to breathe. Hence, in order to breathe and survive, they have to remain conscious, which essentially contradicts sleeping as we know it.
To get around this, dolphins practice a unihemispheric sleep. Namely, only one-half of their brains sleep at a time; the other half stays alert to enable the dolphin to continue breathing and look out for dangers.
They literally sleep with one eye open!
Bonus fact: This lack of ability to spontaneously breathe underwater is probably due to the fact that dolphins are not fish, but mammals – like you and me! Indeed, another famous mammal that moved his habitat, the whale, has to think about every breath it takes and scientists believe that, like dolphins, they sleep with one eye open and one half of their brain awake.