As you probably already know it is highly unlikely a deer or elk would attack, but in the one in a million chance one would (has happened I know). Then Bear spray or mace which is an irritant would just as likely cause them discomfort and to disengage an attack as it would a bear or person that was attacking. Nother words I don't think they'd like it.
Did Trigger attack and you had to use your bear spray? Sorry, I couldn't resist. I would think it would work on deer or elk but who really knows. As mentioned before, it is an irritant and should effect any animal.
I work with horses for a living and it will and does effect horses . It will not necessarily effect their skin but if sprayed in their face wear it is hitting more sensitive like the membranes around the eyes mouth or nostrils it would work. A few years ago in yellowstone a ranger was attacked by a cow elk around Mommouth springs . She had a calf with her and she charged him and knocked him off his bike. He turned his bear defense on it and it turned the cow away .
Over the years I have seen several elk and deer hides left in the woods by hunters and I have to wonder why they do this? I fully understand and agree about getting the hide off the animal as soon as possible to cool the meat, but why not pack out the hide with you and use it? As far as I know there are no state laws that require you to take the hide home, but to me why waste such a beautiful part of the animal? Some might think they have no use for the hide or it costs too much to tan....