Blink or sleep fish

Can you sleep with your eyes open?
No. At least if you define "sleeping with your eyes open" in such a way that you lie in bed with your eyes open all night and still sleep. We humans are not made to sleep with our eyes open. This may also have evolutionary reasons, because there are certainly animals that sleep with their eyelids open or that have no eyelids completely (fish, flies).
Our eyelids protect the eye from drying out. By constantly blinking during the day and by keeping your eyes closed at night. That is why closing the eyelids has purely practical reasons for humans, otherwise the cornea would dry out and eventually peel off. Neither fish have this problem, because their eyes cannot dry out in the water, nor do insects, because they have compound eyes that cannot dry out through the air.
Dolphins and some bird species have developed a completely different technology in the course of evolution. In so-called half-brain sleep, one half of the brain sleeps while the other half of the brain remains awake. Strictly speaking, one cannot speak of "sleeping with the eyes open" because the eye of the sleeping hemisphere is closed.
A seldom occurring phenomenon, which at least gives the impression that you are sleeping with your eyes open, is so-called sleep paralysis. In this, the eyes can open during sleep. But here, too, there is no question of "sleeping with your eyes open".
People who suffer from the symptoms of "lagophthalmos" have insufficient active eyelid closure. As a result, the fissure of the eyelids cannot be completely closed. During the REM phase, the time in which you dream, you can observe natural back and forth movements of the eyes. Affected people therefore unintentionally slumber with "half-open eyes".