There is such a thing as "a living death". It stands in stark contrast to eternal life, in Christ. A living death is being alive without belonging to Christ. Billions of people throughout the centuries have experienced that on earth, even before they die physically. They have lived and died without Christ, and when they die physically, the state they were in when they died continues for eternity because people who die step out of time and enter eternity.
Paul explained this in Romans chapters 7 and 8. There's too much to detail here so I shall just refer to those chapters as they relate to the one verse Jesus spoke in John chapter 17.
Jesus spoke of all the people God had given to him, who were to enjoy eternal life, which Christ would give them (vs.2). They stood in stark contrast to those others who would not get that gift from Christ. Those ones were identified by Christ in chapter 17 as those who hated his people. Why? Because Christ's people "are not of the world, even as I am not of the world. I pray not that thou shouldst take them out of the world, but that thou shouldst keep them from the evil" (vss.14-15).
This means that those who will not get this life eternal in Christ are those who hate those who do have that gift, identified by them not being in the world. Worldly people are not spiritual people who belong to Christ by faith; they are in the world, and they are not kept from the evil. This happens while they are in the flesh, living on earth. They are dead to Christ, even while living on earth.
This is where Paul's explanations are helpful. He spoke of how Christians (including himself) used to live while in the world:
"For when we were in the flesh, the motions of sins, which were by the
law, did work in our members to bring forth fruit unto death. But now
we are delivered from the law, that being dead wherein we were held;
that we should serve in newness of spirit, and not in the oldness of
the letter...
For I was alive without the law once: but when the commandment came,
sin revived and I died... For sin, taking occasion by the commandment,
deceived me, and by it slew me" (Romans 7:5-11).
There you have it - how people who remain in the grip of sin are dead though they live - a living death is their portion, even before their mortal bodies die physically!
Have you heard the expression, "He's a dead man walking"? That is said of persons on Death Row, in prison, those who will be executed. They have been found guilty of crimes that warrant (in that country) execution, and the idea is that they are as good as dead already. That has a chilling application spiritually, with regard to this question.
Paul knew what it was to be under that sentence of spiritual death, due to the sins against God that he had committed. He knew that, no matter how hard he tried, he could not stop sinning and cried out in despair, "O wretched man that I am! Who shall deliver me from the body of this death?" (vs.24) Then he exclaimed that it was Christ who had delivered him from that living death, so that:
"There is therefore now no condemnation to them which are in Christ Jesus, who walk not after the flesh, but after the Spirit... For to be carnally minded is death; but to be spiritually minded is life and peace... And if Christ be in you, the body is dead because of sin; but the Spirit is life because of righteousness" (vss.1-10).
This means that whenever Christ removes that condemnation (sentence of death), they then enter into life eternal, even while in the flesh (John 5:24 & 6:47.) It happens the moment a repentant sinner truly believes in Christ. Conversely, unrepentant sinners are "dead men walking" and that will remain their state when their mortal bodies die; they will step out of time and enter eternity in that unforgiven, condemned state, without Christ. There is not one bit in the Bible that says the wicked will cease to exist. There is no mention of 'soul annihilation'. If there was, nobody would need to ask questions about that.
