While I believe the soul is emergent from and dependent upon the body, I do not believe that we are our bodies. If this were the case then we would not retain our identity as the atoms and cells in our body constantly recycle. Our soul is like a song and our language is the lyrics - our body is the instrument it is played on. It has a certain duration in time, and a certain rhythm and its consistency is what makes it a distinct entity.
Nor is a river the same as the water that flows in it. An old Greek proverb has it that you never step into the same river twice for it is not the same river and you are not the same man. I would argue that a river and a man are a process rather than a material thing and that Heraclitus was wrong. It is the same river and it is the same man, even if it’s not the same water and soil.
Our consciousness, our narrative is a soul. A “natural soul”, dependent on the body.
Yet the body is also dependent on the soul. This is why poor mental health often leads to poor physical health and why there is a placebo effect. A body joined to a soul of poor spirit will usually wither and die quickly.
I do not believe our souls are immortal. I couldn’t rule it out but there is little evidence in favor of this. They seem to be bound to the time between the creation and destruction of our body and brain.
In a sense however you never cease to exist because all time is equally real and the present moment is relative to how much internal time has passed since your mind was formed. Additionally your actions in your life will influence events until the end of time in ways that are unknowable.
I suspect it likely that our descendants will become liken to gods and will live much longer and be much wiser than we are. And who knows, they may even escape this Universe and cheat entropy, attaining true immortality.
Perhaps they even mastered time travel and in fact created the Universe, in a paradoxical causal loop.