Human beings do have a spiritual component to our makeup. As Job 32:8 says, "There is a spirit in man." Zechariah 12:1 tells us that God "forms the spirit of man within him." And the apostle Paul pointed out, "For what man knows the things of a man except the spirit of the man which is in him?" (1 Corinthians 2:11).
It is this human spirit that imparts human intellect to our physical brains, creating the human mind. This is what makes people vastly more intelligent than animals.
Yet this spiritual aspect of human existence is nothing like the immortal soul concept. It is something distinctly different. The spirit in man is not animate of itself. It is not a spirit entity that "lives on" after death. As Scripture shows, the human spirit has no consciousness apart from the body, for man is mortal. When we die, we will have no awareness of anything at all.
Ecclesiastes 12:7 tells us that, at death, "the spirit returns to God who gave it"—where it is retained until the future time when God places those individual spirits within new bodies at the resurrection, thereby bringing individuals back to life with their personality and memories preserved and intact.
The human spirit is critical to our destiny, since God's Holy Spirit joining with it is what makes us God's children (Romans 8:16). And just as the human spirit gives us human understanding, so God's Spirit gives us higher, godly understanding (1 Corinthians 2:10-16). We are not born with the Holy Spirit but receive it from God following repentance and baptism (Acts 2:38).