Good question Mila. The short answer is yes! There are ways to test if an animal sleeps: They must 1) show a time of prolonged inactivity, 2) during that time have a reduced startle response, 3) if you deprive them of sleep, the next time they sleep it is for a longer period of time.

I have seen a fish sleep in the ocean while scuba diving and accidentally woke them up! I research a species of fish that live in caves and they have evolved to sleep very little compared to their cousins that live in rivers. Sleep is a fundamental to life. Even flies, worms, and jelly fish sleep. The crazy thing is, we don't know exactly why. Maybe you can figure that out some day!

I consulted with a scientist who scuba dives to research fish behavior. She says that she has seen fish resting, but we don’t know whether they sleep in the same way as humans and other land mammals. For one thing, fish don’t have eyelids! This lets them stay alert to predators while still conserving energy. Some fish also hide in sand or under rocks while they rest.

Sleep has been observed in zebrafish, a model organism commonly used in biological research. In fact, Mutant zebrafish that lacks certain peptide receptors exhibits insomnia. Furthermore, sleep has been observed in silk moth, fruit fly, and roundworm.

Fish seem to 'sleep' more like we rest. They don't have eyelids, so they sleep with their eyes open.

In aquariums, fish sleep cycles are often determined by interior lights—the fish will sleep when the lights are turned off. ... Take zebra fish, which naturally drop their tails low and settle in at the bottom of tanks or just beneath the surface of the water to sleep read here.

Do Fish Sleep with Their Eyes Open?
Wonderopolis. "The simple answer is yes! They are sleeping, and they can sleep at any time during the day or night. Fish do sleep with their eyes open, because they don't have eyelids (except for some sharks) to close!"

We might be able to say that fish sleep, but their sleep state is not the same as ours. It depends how we define the state of sleep. For instance, birds and mammals sleep with their eyes closed, and their brains have some distinct patterns during sleep, but fish do not have eyelids or the part of the brain with those distinct patterns. However, if we look at behavior, we can see that fish are not always awake. Some fish do not move or respond to stimuli at night, but this motionless, unresponsive state may not always be present. For some fish, this state may disappear during migration, parental care to their young, etc. Some fish, such as zebrafish, have a somewhat constant amount of sleep every day, and will respond to sleep deprivation by sleeping more the next few days.

There are certain fish that seem not to sleep, such as tuna and some sharks. As you can see, this question does not have a simple answer and still needs more research.