The concept of everlasting life is referenced a lot in the New Testament, and along with the aforementioned concept of a bodily death being different (and less significant) than the kind of death God could put on you, these point towards an extra essential part of us that isn't physical.

Add to that descriptions of us being raised with bodies unlike what we have now: what makes those bodies us if they don't share the same physical nature as what we have now -- surely there is more to us than a physical body in a complex and constant chain reaction if "we" can be enjoy everlasting life after the death and decay of our bodies.

The Bible doesn't appear to describe what a soul is, but I don't think it needs to. My soul seems intuitive to me, and is not something I had to be taught. Instead, "soul" is a word that I associate with something I've always had based on the context in which it is often used. Apparently Jewish tradition does contain descriptions of the soul, its function, its origin, and so on. But I don't know anything about the details of that.

So, yes, the idea of an immortal soul is supported by the Bible, even if we aren't told, "This is what a 'soul' is".

Jibb