Your “spirit” is nothing more than the infinite essence that permeates all existence, whilst temporally occupying the brain-body space.
In actuality, the physical body is our soul. Granted that the body is truly immortal, whereas the “dweller” within it is ephemeral. When you “die” the body will corrode into the earth, and this corrosion is merely a transmutation from a single “body” into many. The body decomposes and from its decomposition comes a myriad of life-forms, other organic matter and minerals which are born from the earth. This makes the body immortal, for it has no real beginning, thus it has no end.
Life is eternal transformation.