Some do. I used to have a chinchilla that slept with his eyes open but my other one sleeps with his eyes open.
No they don't.
Yes almost all farrets sleep with eyes open or partially open
Yes they sleep with their eyes open :)
No. Kookaburras are not known to sleep with their eyes open.
No. They close their eyes like us when they sleep.
yes they sleep, but there eyes are open
No, they close their eyes.
Some people can sleep with their eyes open. Many 'sleepwalkers' give the appearance that they're fully awake - including having their eyes open.
open
no
no
No.
Possums usually sleep with their eyes closed, but you may see the occasional possum asleep during the daytime, with its eyes open.