What is the basis in the Scripture for the doctrine that those who haven't received salvation during their physical life, especially those who heard of Christ and yet chose not to pray to Him, will still have a chance to get saved after their physical death?
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The only passage I am aware of is 1 Peter 3:18-20a:
It should be noted that this is rather a "standalone" passage, in that there is nothing similar in Scripture, so it is unwise to base a doctrine on it. However, you could read into this:
Note though that this is speculation and not a central belief of any mainstream Christian denomination (that I know of - but see @brillant's comments, below).
Judas not only prayed for the dead men ("turned to supplication"), he also made a sin offering on their behalf. It is also the first reference of the resurrection in the scriptures.
Keep in mind that this is considered Scripture in Eastern Orthodoxy.
One more place in Bible which is (as far as I know) 1Corinthians 3:15. Here I post it with some context (1Cor 3, 10-15).
It definitely talks about those who build on Christ and will be saved, but it recognizes two categories: those who recieve reward and those who will be saved too, "but only as one escaping through the flames". Catholics interpret this as a reference to fate of those saved after death: they will go to heaven either directly (the reward) or through purgatory (the flames). Or through "something like purgatory" - I think Eastern Orthodox doctrine teaches something like this.