Nuttall's Cottontails can sleep with their eyes open! This makes them look awake and less vulnerable to predators, and they can more readily detect danger when resting.
To make sure their eyes don't dry out while they sleep, they will routinely blink their "nictitating membrane", or clear third eyelid. This third eyelid is present in most other animals, including birds, fish, reptiles, and most other mammal species. Humans and other primates are oddities in that our nictitating membrane is nonfunctional has almost completely disappeared!
The Bald Eagle pictured below was in the middle of blinking its nictitating membrane as the photo was being taken, so you can see the membrane covering half the eye.
(Credit: Jörg Hempel via Wikimedia)