Does Buddhism cause longevity? In this book about Bodhisattvas it says that  traditionally Buddhism can cause longevity in a way that parallels the longevity that is sought after by Taoism. This kind of longevity arises in the same way that the psychic powers do i.e. it is a natural result of practice but it isn't the end point and it isn't something that should be grasped after. This is the first time I have heard of Buddhism in connection with longevity. Has anyone got any further details about this and perhaps some textual references where this is referred to? 

Please note - I'm not interested in scientific studies or connections to well-being/stress reduction and similar in this question. I want to focus on the texts and traditions rather than more modern studies.

Many Thanks
Quoting from Maha-parinibbana Sutta: Last Days of the Buddha,


  
  And the Blessed One said: "Whosoever, Ananda, has developed, practiced, employed, strengthened, maintained, scrutinized, and
  brought to perfection the four constituents of psychic power could, if
  he so desired, remain throughout a world-period or until the end of
  it. [21] The Tathagata, Ananda, has done so. Therefore the Tathagata
  could, if he so desired, remain throughout a world-period or until the
  end of it."
  But the Venerable Ananda was unable to grasp the plain suggestion, the significant prompting, given by the Blessed One. As though his
  mind was influenced by Mara, [22] he did not beseech the Blessed One:
  "May the Blessed One remain, O Lord!. May the Happy One remain, O
  Lord, throughout the world-period, for the welfare and happiness of
  the multitude, out of compassion for the world, for the benefit, well
  being, and happiness of gods and men!"
  And when for a second and a third time the Blessed One repeated his words, the Venerable Ananda remained silent.
  Then the Blessed One said to the Venerable Ananda: "Go now, Ananda, and do as seems fit to you."
  




So the above is a textual reference.

Off-topic to the question, this (i.e. not having asked the Buddha to remain) is a reason why Ananda was reproached at the first council after the Buddha's death. Later in the same sutta Ananda realizes what the Buddha had been saying and asks him then (asks him three times), but the Buddha says no:


  
  And the Blessed One answered, saying: "Enough, Ananda. Do not entreat the Tathagata, for the time is past, Ananda, for such an entreaty."
  


and,


  
  "So also at Vesali, Ananda, at different times the Tathagata has spoken to you, saying: 'Pleasant, Ananda, is Vesali; pleasant are the shrines of Udena, Gotamaka, Sattambaka, Bahuputta, Sarandada, and Capala. Whosoever, Ananda, has developed... Therefore the Tathagata could, if he so desired, remain throughout a world-period or until the end of it.'
  
  "But you, Ananda, were unable to grasp the plain suggestion, the significant prompting, given you by the Tathagata, and you did not entreat the Tathagata to remain. For if you had done so, Ananda, twice the Tathagata might have declined, but the third time he would have consented. Therefore, Ananda, the fault is yours; herein you have failed.
  "Yet, Ananda, have I not taught from the very beginning that with all that is dear and beloved there must be change, separation, and severance? Of that which is born, come into being, is compounded and subject to decay, how can one say: 'May it not come to dissolution!' There can be no such state of things. And of that, Ananda, which the Tathagata has finished with, that which he has relinquished, given up, abandoned, and rejected — his will to live on — the Tathagata's word has been spoken once for all: 'Before long the Parinibbana of the Tathagata will come about. Three months hence the Tathagata will utterly pass away.' And that the Tathagata should withdraw his words for the sake of living on — this is an impossibility.