I am an animal, and I am at least intermittently self-aware, so yes.

But I'm guessing you wanted to know whether non-human animals are self-aware. We could spend some time trying to sort out exactly what counts as self-awareness, and that would be a lengthy though worthwhile exercise. But the short and plausible answer is that some are and some (perhaps most) aren't. One reason to think that some are comes from research with mirrors. Some elephants and some chimps, at least, seem to be able to figure out that what they're seeing in a mirror is their own reflection. You can read a short account of some of the research here.

It is true that a number of psychologists treat intelligent use of mirrors as evidence of self awareness. But I am not convinced. Animals can gather information about their own bodies via various forms of perception, including, of course, vision. Some can also use a mirror - extending the range of their vision - to get information about their own bodies. But I don't see how that implies that they have any concept of self. My guess is that lots of animals do have something that we might reasonably call 'self awareness'. But I don't know of any serious evidence for this.