Do we have [a] soul?
Has science proven it?
Depends on what the word 'soul' means. (Isn't that annoying? It's how philosophers typically address questions.)
In the ancient world people understood 'soul' (or rather the equivalent words in their languages) to mean something like 'animating principle', whatever it is that gives life to living things. They didn't always agree on what that was.
Aristotle argued that the soul was simply the form of the body, in other words how the body behaves, and not something that can exist separate from the body. Modern physiology and molecular biology have expanded on that idea tremendously, and have gone a long way towards describing and explaining how life operates in the Aristotelian functional sense.
But most of the ancients imagined that the soul, the life-principle, was some kind of additional substance, a supernatural or even divine stuff whose presence animated flesh and made it sentient and alive.
It was often identified with the breath. What animated life was the presence of a vitalistic gas of some kind, something that could be breathed into material bodies to animate them and that those bodies breathed out again at death. That's why ghosts are traditionally pictured as spirits or vapors. We even see alchemists using crude chemical distillation apparatus in hopes of isolating these animating spirits.
Some of the Platonists, the patristic and medieval Christian theologians, and Descartes and much of modern philosophy that subsequently followed, have imagined 'soul' as some kind of immaterial stuff. Soul substance was a separate realm of substance distinct from and contrasted with physical substance. These kind of thinkers were more inclined to identify soul with mind, with the principle of human subjectivity, than merely with whatever made living bodies move. They imagined soul as only temporarily associated with material (in Descartes' view clockwork) bodies and fully capable of existing perhaps forever in its natural state, independent of matter.
The latter-day stages of these disagreements are still being fought out as we speak in the contemporary philosophy of mind, between those who seek to understand the mind in physicalistic functional terms, and those who argue that mind, consciousness and qualitative awareness ('qualia') can never be reduced to physicalistic terms and must therefore be a separate realm of being.