The Bible does indeed teach that every human being has a spirit, but that spirit is neither immortal nor a soul.
I. The Spirit in Man
We discuss the biblical concept of the "spirit in man" extensively in our free booklet, "The Theory of Evolution--a Fairy Tale for Adults".
In this Q&A, we are quoting the following excerpts:
In our last Q&A (in Update #438, dated April 15, 2010), we explained that God gives every person, apparently at the time of conception, a "spirit," which the Bible calls the "spirit in man." This spirit is not a soul--the person is the soul--nor is it a conscious being. When the person or the soul dies, the spirit in man returns to God who gave it (Ecclesiastes 12:7). It is being stored in heaven until the time of that person's resurrection.
"You are the light of the world. A city that is set on a hill cannot be hidden."-Matthew 5:14