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My answer to this question will have to be short. The reason for this is, as stated, I'm not exactly sure what the question means. One first needs to define the term “soul” before he/she can consider whether an animal possesses such a thing. In fact, the term “soul” isn't even well-defined for humans, so answering this question is, a fortiori, more difficult.
I have a suspicion, though, that what is intended is some immaterial substance, which animates the animal and survives death. If this is the case, then I think the answer is unequivocally no. This won't be a popular answer, but there is absolutely no reason nor evidence to suggest that this is the case beyond human sentimentality and emotional attachment.
