It is a fact that the brain operates on chemical processes. Anyone with the tools and the knowledge can test this and find it to be true.
It is a fact that a neuron is the means through which mentioned chemical processes work to relay and interpret information. Again, this is testable and true every time.
It's easily provable that reasoning, memory, and even self-awareness are accomplishable by this complex set of electrochemical-neuron workings. There is nothing to show for the metaphysical concept of a soul, however. While it is not a fact that a soul does not exist, there is very little to support its existence outside of culture myths. One can easily say that organisms are warm because they have sacred, literal fire in their heart; it justifies the end result (the fact that organisms are warm), but as an explanation itself it does not hold up to modern scrutiny. If one is referring to a soul as simple self-awareness and not some magical idea that transcends the body, then yes, I guess we could say we and a select few animals have souls. If one is referring it to as a religious concept, I don't really care.