Never seen them with eyes closed full out asleep but I have seen antelope fully asleep in the middle of the day, you can walk right up to them.

A couple years ago in Illinois I was sneaking into my stand in the pitch-black of morning. I was intentionally going in extra early because I wanted to beat the buck I was hunting into his bed-room. I was quietly sneaking through the timber without the use of a flashlight because I could see well enough to navigate a path I knew well.

Without warning, a deer exploded from what seemed like 2 ft. away from me. I nearly soiled myself on the spot - it scared the hell out of me! This deer had to have been sound asleep, and didn't wake up until I was right on-top of it. Generally though, I think deer take cat-naps throughout the day, rarely falling into a deep sleep. I read an artical in D&DH about this same topic last year sometime. They had some pics of some bucks with they're eyes closed, and using they're racks as "kick-stands" for they're heads. Some cool pics if I remember correctly!

I have had a doe bed down in-front of me before and lay completely out on it's side, head down, the whole nine yards. Watched her change positions several times throughout the day and everything. Legs stretched out to the side like it didn't have a care in the world! This is the only time I've seen this though, generally they are bedded with they're heads in the up-right and ready to flee position!

Last year, I had a couple twin fork-horn bucks by my stand. They were banging their heads together and making some rubs for a while. After about half an hour, they got tired of that and laid down foranother half hour or so, about 15 yards from my stand. They both had their eyes closed for various lengths of time, probably not longer than 5 minutes at a time. Definitely asleep.
Then they got up, and walked away.
Short of shooting a deer, by far the coolest hunt I have ever been on as far as seeing things I've never seen before go.

Of course they do. They are an animal, right?

Nice pics hardcore. I guess that settles that.

Wow, I never imagined that this question would result in so many replies.
I would like to thank everyone for some very interesting and informative feed back.
Hardcorehunter - those are great pictures, I have never seen a deer that relaxed before. Even when I brought the kids to the Catskill Game Farm (local zoo) the deer there would beresting but never sleeping like your pictures.