Does a man before being saved possess spirit?

Does a man before being saved possess spirit? As I read messages by Spurgeon, this question came to my mind. I have been searching for an answer for few weeks and I think I found an answer now.

Genesis 1:26 reads:

And God said, Let us make man in our image, after our likeness: and let them have dominion over the fish of the sea, and over the fowl of the air, and over the cattle, and over all the earth, and over every creeping thing that creepeth upon the earth.

Genesis 2:7 reads:

And the LORD God formed man [of] the dust of the ground, and breathed into his nostrils the breath of life; and man became a living soul. 

In Genesis 1:26, God made man in "our image, after our likeness". The teaching from the church has been that man is made in God's image refers to the fact that man has spirit as God is spirit and that the spirit of man died when he sinned as God had said in Genesis 2:17,

But of the tree of the knowledge of good and evil, thou shalt not eat of it: for in the day that thou eatest thereof thou shalt surely die.

and man died spiritually after he sinned as seen in Genesis 3. I also used to think that the breath of life was God's spirit and that that is how man came to possess spirit.

However, breath, Strong's Number H7307 or the Hebrew רוּחַ (ruwach ), is used in Genesis 7:15 as well to indicate the breath of life in animals. Genesis 7:15 reads:

And they went in unto Noah into the ark, two and two of all flesh, wherein [is] the breath of life.

So the breath of life cannot be spirit since animals do not possess spirit.

Then, what does it mean that man became a living soul? Rather than saying a living soul here means a living spirit, we just have to accept that man was first formed with the dust of the ground and now possessed a soul. So an unregenerate man possess a flesh and a soul but not a spirit.

1 Corinthians 15:44-45 read:

It is sown a natural body; it is raised a spiritual body. There is a natural body, and there is a spiritual body. And so it is written, The first man Adam was made a living soul; the last Adam [was made] a quickening spirit.

Here "natural", Strong's Number G5591, matches the Greek ψυχικός (psychikos), and "soul", Strong's Number G5590, matches the Greek ψυχή (psychē). So these two words have the same root and "natural" means "soulish".

Spurgeon writes in The Soul Winner:

We are first in the natural or soulish stage of being, like the first Adam, and then in regeneration we enter into a new condition, and we become possessors of the life-giving "spirit."

Does a man before being saved possess spirit? The answer seems no or at least the unsaved man is soulish and not spiritual.