If you venture to a local pond/lake and observe the mallards and black ducks, some of them have one of their eyes open while sleeping because they can keep half of their brain awake. I just happened to notice it while on a daily walk and had to go look it up afterwards. Ducks also make outrageously loud quacks when alarmed (sounding similar to a single person laughing too hard at an unfunny joke). Made me jump more than once when walking after dark.
Not all of them will always sleep with one eye open, but at least a few in a group will in order to keep watch from coyotes, birds of prey and stray cats when they're not in the water.
Any animal without big selection pressures to develop and maintain this ability seems to lose it rapidly, as other adaptations of the brain for other purposes seem to preclude USWS. Only animals which really need it seem to have it.
I know first hand someone who have driven several kilometres only to "wake up" and not being able to explain. This is of course extremely dangerous and in every case I know of it has ended with a damaged / wrecked car though luckily no injuries or fatalities. (Back country roads, middle of the night etc).
What happened in at least one of these cases was dreaming with eyes open. From what I hear that might not be to uncommon but here I'm back to old wives tales.
There is a case to be made with these situations that you weren't actually asleep.. you just didn't lay down memories for the period of time, because nothing happened.
You feel like time is missing because you have no recollection of that time - but that does not have to equate to you not being awake, or able to respond.