Did you mean to ask "do they have a Spirit"? Your soul is your mind and emotional realm of your body. Your spirit is what lives on after you die.
No, meant to ask about the soul. I would disagree with your assessment of the distinction between spirit and soul. The tradition I belong to holds the belief that the body and the soul will be reunited when the world is reborn, so we see no distinction between the two.
Either way, what I was intending to ask is, "Do you believe that animals have an entity that endures after natural death?" Call it what you will.
I'm enjoying the responses. And don't think that I've got some sort of "zinger" saved up here. I don't. I have some thoughts, but no solid answers.