We are 'spiritual' creatures - but not possessing immortal life until resurrected out of this age. Until that change takes place, we are predominately flesh and the human spirit in us is to provide a faculty for reason and intellect etc.
Again we see the contrast of the man's spirit with inherent free-will component - able to rebel and sin, contrasted against the spirit of God that would transform him eventually to Himselfgodliness - being saved.
So, the spirit in man is of God, but by no means divine - NOT holy by any means. Just as the spirit in animals is not holy either - only difference is they are not being saved or redeemed.