Actually, Adam and Eve were considered immortal before falling. The warning from God that if they ate of the tree of knowledge of Good and Evil, they would die, implies, that they would not die if they never disobeyed and ate from that tree. While we were created lower than the angels, we shared in that expectation of immortality before falling.
The eternal life that we gain by placing our trust in Jesus, is not something that we can lose. As Paul said, "Rom 8:38 For I am persuaded, that neither death, nor life, nor angels, nor principalities, nor powers, nor things present, nor things to come, Rom 8:39 Nor height, nor depth, nor any other creature, shall be able to separate us from the love of God, which is in Christ Jesus our Lord. "
I follow that Adam and Eve had body, soul, and spirit prior to the fall, but after the sin, they became separated from God in spirit. Death being a state of existence involving separation meaning they no longer had fellowship with God spiritually. They were still alive physically, and soulishly as they recognized they were naked, but without the action of God they did not have spiritual fellowship with Him.
Note also, that after eating from the tree of knowledge of good and evil, God proceeded to post an angel lest they eat from the tree of everlasting life without appropriate salvation.