In theory yes but I don't think we have done the experiment. That said, some of the animal models we use for researching depression would likely apply to any animal. For example the mouse forced swim test is used as a depression test. The mouse is put in water and swims till it gives up, then give it an antidepressant and that mouse will swim longer before giving up. You might say "is this really a model of depression?" Well since we can't ask animals if they are depressed things like this are the best we can do to "depress" the mouse. It may well not be a great model but stuff like this is what we have. Note they do not drown the mice, just record how long the swim before giving up, then they are taken out. I would guess you can do the same experiment with an elk, and it is quite possible the antidepressants you give it might make it swim longer. Which as far as we know, that elk gets to depressed to continue and the drugs make him less depressed and continues longer. But again, we can't ask the elk, so we can't be certain just like the mice.

As to whether animals get depression like humans we can't really say again since we can't ask them. However a lot of animals have evolved to be very anxious due to predators. You may notice deer are very skittish. This evolved for a very good reason, a non anxious deer is likely to more easily to be killed by a predator. Now is this anxiety disorder in an animal? Probably not, but they certainly seem very anxious all the time. And I am betting if we gave them benzodiazepines in the wild they would be a lot less anxious (to their determent). If I were to guess it would not surprise me if animals get depression as a condition because we humans certainly did. We evolved with it and the question is why? There are theories that depression may have been an adaptation to maintain bodily energy in a limited food environment perhaps or a consequence of some immunological factors that we needed but resulted in the side effect of depression. None of those theories are proven but they do make some arguments why depression remained with humans when you would think it would be selected against over time.