I am of the view that 'the eternal life' is a reality, and the central concern of all of the various spiritual faiths and philosophies. It is obviously central to the New Testament, but it is also represented in the Pali Canon, in passages such as these:
I have heard that on one occasion the Blessed One was staying in Savatthi, at the Eastern Gatehouse. There he addressed Ven. Sariputta: "Sariputta, do you take it on conviction that the faculty of conviction, when developed & pursued, gains a footing in the Deathless, has the Deathless as its goal & consummation? Do you take it on conviction that the faculty of persistence... mindfulness... concentration... discernment, when developed and pursued, gains a footing in the Deathless, has the Deathless as its goal and consummation?"
But 'the deathless' does not refer to the life of 'the person'. It is not as if the person, or the personality, is preserved for an infinite duration of time. There is a very real sense in which the 'eternal' is outside of time, not 'an interminable period of time'. This is altogether impossible for the mortal mind to envisage or imagine, however this is the realm that the Buddha has entered into.
Nirvana, for which 'The Deathless' is another name, is the extinction of the ego. This doesn't mean simple non-existence, but higher being. This is why the Buddha is called 'Tathagatha', 'gone thus'. That which is eternal is 'the supreme identity', the immortal spirit which is our real being - the buddha-nature, in some Buddhist traditions. Through discernment, as the Buddha says, this being can be realized, and when realized, then there is liberation from death through attaining an identity that is greater than that of the mortal body.
This is also expressed in Christianity, when Jesus said: 'He that looses his life for My sake will be saved' [Matt 16:25]. This was also exemplified in His own self-sacrifice.
You might say 'but this is a type of impersonal monism'. But just as the eternal is revealed in the particular, so the spirit is made real in the form of one who has transcended the ego. This is the universal message of the spiritual philosophies of the world.