Everything we know about the brain can be show to be a physical process. There is nothing about the brain that can be shown to be a non-physical process. Therefore the only reasonable argument is that there are no non-physical processes necessary for the brain to function.
There exists nothing to support any other argument.
You are assuming that the brain is the mind. This is an unsubstantiated assumption. In spite of the correlations which neuroscience has observed between areas of the brain and aspects of consciousness, there is no sound basis for reductionism. Even if we accept as axiomatic (and there is no consensus on this, either in science or philosophy) that consciousness emerges from electrochemical activity in the brain, we must still acknowledge qualitative experience as a clearly identifiable, unique phenomena in it’s own right.
And then there are those - scientists as well as philosophers - who consider that consciousness may be a fundamental property of existence. After all, the
only thing we can observe with absolute assurance is, as Renee Descartes pointed out, that we are conscious.
“Consciousness cannot be accounted for in physical terms. For consciousness is absolutely fundamental. It cannot be accounted for in terms of anything else.”
- Erwin Schrodinger