So the serpent told Eve the truth and God lied to her.
A rather timely example occurred recently in my life. I was at a funeral on Thursday. My cousin died on Sunday, but her body did not cease to exist at that time. On Sunday, her spirit departed, leaving her body in a state of death. Of course, the body continued to exist, as it was brought to the mortuary for preparation for the funeral. According to the Bible, her spirit continues to exist, and her spirit may be alive (Heaven) or dead (Hell). That is of course, dependent upon whether she put her trust in the Lord, or not.
Spiritual death means that one's spirit exists in a state of death, which is the absence of the Lord's presence in your spirit, but it is NOT the spirit's ceasing to exist. When God spoke the truth ("...surely you will die." Gen 2:17) and Satan, that old serpent lied ("...Surely you will not die! Gen 3:4), it was concerning the spirit.