This depends on your theology on two different points: whether it is a soul that makes man alive, and whether saying that man was created in God’s image entails that man has a spirit (or soul). On the first question, we are told that God breathed the breath of life, and thus Adam became a living soul. Now, these particular English words, from the KJV, are not dispositive, because the original words are perhaps not as clear. “Breath” translates נְשָׁמָה or neshamah, which is most typically breath, but is closely connected to “spirit,” so for example Job 27:3 has “All the while my breath [neshamah] is in me, and the spirit [ruach] of God is in my nostrils”. Ruach (רוּחַ) means not just spirit but “wind”; the suggestion is both that God’s breath (neshamah) is like the wind (ruach), and also that the principle of life is something invisible, like (but obviously not literally the same as) the wind. The word ruach was also used in Gen 1:2, in the phrase “the Spirit of God.” This leads us to the second question: the image of God is some sort of similarity, and we are not precisely told how we are similar (see on Gen 1:26-27). The text might well be thought to suggest that man is like God, or represents God, by exercising dominion over the earth. God’s ultimate act of dominion, namely creation, was done through the creative Spirit. So it is reasonable to think that the original author and readers of Genesis would take the text to imply not just that Adam was a living being, but possessed an active, “ruling” spirit (one capable of “dominion”), akin to God’s, but acting in a much more restricted sphere.