I can easily float on my back, and breathe normally without my mouth or nose going below the water. Some people cannot do this and sink below the water when they breathe out. I know that the body paralyzes the muscles during REM sleep, at least I think it’s REM sleep. Would you drown floating on your back; assuming that the water is completely still, and you do not move much when sleeping?
People lost at sea have reported doing this. They probably weren’t in a REM sleep state. In salt water they were more buoyant, so that might make it more possible. In any sleep state you’re likely to wake up if water starts going down your throat.
I find it very hard to believe that you can sleep for more than a few minutes without moving your head at all. And once you move it, you disturb the equilibrium, and you’ll get water in your nose the next time you inhale.
No way I can test any of that, because I sink like a rock in water, and I can’t sleep on my back even in a bed.
I once fell asleep in a sensory deprivation tank while i was floating. The water had kilos and kilos of epsom salts so floating was a given.
I remember two ads in Time or some other weekly way back for “best possible nap.” There seemed to be a lot of inventiveness back then. One was the highly saline tub mentioned above. The sleeper was kept upright by currents running underneath (not sure how they fixed the head turning.)
The other invention was a big opaque plastic inflatable canopy which the sleeper crawled into, lay down, and was drummed with a soft rhythm that mimicked being inside one’s mother’s womb. There was even a choice of supplementary sound (music, nature, ego-boosting pitches, etc.) The article was titled “A nap by any other name.”
This guy certainly thinks so.
As does one of the people in the Comments section.
Go to the Dead Sea. It’s easy there. 
Well, there’s this story about a Marine who fell off an aircraft carrier back in 1995 and spent the next 2-and-a-half days floating in the ocean. Somehow, he was able to catch some zzz’s:
I have noticed since I have gotten older I am more bouyant and can fall asleep in the ocean while floating on my back, never for more than a few minutes or seconds but I have dozed off many times.