Your soul (consciousness) is an emergent property of your brain activity.
The way you’ve framed this question doesn’t take into account emergent properties. It’s not a question of “is the soul part of your body, or something entirely separate,” because with emergent properties it’s not an either/or. Rather, the soul exists on a different “level” than the behavior of its individual neurons.
Many animals are obviously conscious. Extremely simple animals, like bivalves, are not, because they don’t have nervous systems. I don’t think consciousness is this binary either/or thing either. Rather, I think an organism can have varying “amounts” of consciousness, depending on how complex its nervous system is. Humans and chimps have highly complex brains. Lizards? Not so much, but they still feel pain and have a sense of self. Starfish? It’s an unclear boundary.
Souls do not exist after life, because they emerge from brain activity. No brain activity, no soul—for the same reason that if you smash your computer you won’t get to run your computer’s programs anymore.