This is no trivial point. It is essential that we understand that Jesus was completely God in every way, and completely man in every way. The atonement demands it. Only if He was a true man could He be our Kinsman Redeemer and the Substitute for our sins. If Jesus Christ did not have complete humanity, then we don't truly have the Biblical doctrine of the Incarnation nor atonement for sins.
Man is tripartate.
He has a body, soul and spirit. The body is the visible part. Thus, Jesus here gave up the life force which kept His body alive. The soul, however, was a different matter. Since the soul for a man is his true persona, this must have been nothing less than the Spirit of the Almighty God, for the Person of Jesus Christ is God Himself.