This is an interesting question, because the answer has a lot to do with your specific beliefs.
Atheists: Of course not, nothing has a soul (i.e. a spirit that lives for ever).
Creationist A: Yes, the Bible says so (quotes passage).
Creationist B: No, the Bible says so (quotes passage).
Now where it gets interesting is in theological evolution. Assuming this starting point, humans evolved from our ape ancestors. If humans have souls and other animals don't, at what point did human ancestors start having souls? If animals have souls, what is the nature of their soul? Did our ape ancestors have souls? Our reptile ancestors? Our fish ancestors? How about the sponge, or the first eukaryotes?