No, the undying, immortal soul is not mentioned in the Bible. The word "immortal" is found only once in the Bible, and it is in reference to God (1 Timothy 1:17). (more on this later)
What was Eve told in the garden and how did the serpent (Lucifer) answer?
3 But of the fruit of the tree which is in the midst of the garden, God hath said, Ye shall not eat of it, neither shall ye touch it, lest ye die.
4 And the serpent said unto the woman, Ye shall not surely die: (the original lie of immortality and the decision to obey Lucifer and not God))
5 For God doth know that in the day ye eat thereof, then your eyes shall be opened, and ye shall be as gods, knowing good and evil.
The Bible uses the word “immortality” only five times, and the word “immortal” only once. In this lone instance the term is applied to God. “Eternal, immortal, invisible, the only wise God.”
The five references that contain the word “immortality” are as follows:
Romans 2:7. In this text the Christian is exhorted to “seek” for immortality. Why should he seek for it if he already possesses it? In this same book of Romans, Paul quotes the prophet Elijah as saying of his enemies, “They seek my life.” We understand from this that the prophet’s enemies did not yet have his life in their hands. Therefore, when we are exhorted to seek for immortality, for a life that knows no end, we must conclude that we do not now possess such a life.
2 Timothy 1:10. Here we learn that Christ “brought life and immortality to light through the gospel.” The only deduction from this is that so far from immortality’s being a natural possession of all men, it is one of the good things made possible through the gospel. Paul wrote, “The gift of God is eternal life through Jesus Christ.” Rom. 6:23. Why would we need this gift if we already had undying souls?
1 Corinthians 15:53. This passage tells when we shall receive immortality. The time is “at the last trump.” Then “this mortal must put on immortality.” Why should the apostle Paul speak of our putting on immortality at a future date if we already possess it?
1 Corinthians 15:54. This verse simply adds the thought that when “this mortal shall have put on immortality, then shall be brought to pass the saying that is written, Death is swallowed up in victory.”
1 Timothy 6:16. Here we learn that God “only hath immortality.” This final text settles the matter as conclusively as words could possibly do, and explains fully why we are exhorted to “seek” immortality, and why we are told that immortality is something that is to be “put on” “at the last trump.”
Not only do we learn from these texts that we do not have immortality, but also we are told that God alone has it.
... folks, for some people it may not be comforting to know that when Mom or Grandma die they simply decay, in the ground, "asleep in Christ", waiting for the 2nd Coming, to "put on immortality"; however, that is biblical teaching! It makes no difference, BTW, if you bury or cremate, or are lost at sea; God is quite capable of reconstructing you.
Our comfort is in Christ and our belief in Him, and in the coming day when we meet Him!
But Tiger, I've talked to my deceased mom through a psychic medium! She tells me everything's beautiful in heaven!
Ahh! We'll talk about that later! - Tiger
Additional Reference on the Pagan Origins of the Immortal Soul: