It is settled pretty strongly that life is given by Ilúvatar and the Gift Of Ilúvatar to Men is death. And yet, we see the Nazgûl (and the Dead Men of Dunharrow) refusing the gift and have some sort of existence for an extremely long time. Is there a canonical answer on how could this happen?
It is clear that the Ainur have the power to delay the Gift of Men. There are numerous examples. After Beren's death, his soul resides long enough in Mandos for Lúthien to make a plea on his behalf, and the pair are restored to life. Their descendants are allowed to choose whether they will take the Gift or remain tied to Arda as Elves.
Sauron, as a much lesser being than Manwë, probably lacks the power to change whether a given being is Man or Elf. (Lúthien, most notably, married into Humanity). However, he does have the power to extend the lifetimes of Men far beyond their normal span. For the Nazgûl, their lives seem to be extended essentially indefinitely, although it is an open question whether they would still have died eventually, had their rings not lost their power. But the Ringwraiths are still Men, and with the destruction of the Ring, they died and eventually passed beyond the circles of the world. In one of his letters, Tolkien suggested that Sauron could have recreated the Lord of the Nazgûl after he was slain, but that was only possible while the power of the Ring lasted.
The Dead Men of Dunharrow were cursed to remain by an even lesser power, a king of the Dúnedain, but Isildur apparently had enough ability to delay the departure of the oathbreakers as well. When their oath was fulfilled with the capture of the corsairs' ships, they too departed from Arda.
There are other possible examples as well—the barrow-wights, for instance. What is clear is that magical power can be used to delay the Gift of Men. This can mean keeping men alive beyond the normal span of their years (as with the Nazgûl, or even the long-lived Númenóreans). It can also mean that the souls of the departed can be prevented from leaving Arda immediately after death, although not forever. (This happened to Beren, the Dead Men, and possibly also the Nazgûl.)
The Nazgûl and Men of Dunharrow didn't really refuse Ilúvatar's gift of death; rather they were cursed and didn't have a choice.
The Silmarillion, in Of the Rings of Power and the Third Age, goes into detail about the Nazgûl's predicament:
Men proved easier to ensnare. Those who used the Nine Rings became mighty in their day: kings, sorcerers, and warriors of old. They obtained glory and great wealth, yet it turned to their undoing. They had what seemed to be unending life, yet life became unendurable to them. They could walk, if they would, unseen by all eyes in this world beneath the sun, and they could see things in worlds invisible to mortal men; but too often they beheld only the phantoms and delusions of Sauron. And one by one, they fell under the thralldom of the ring that they bore and under the domination of the One Ring, which was Sauron's.
Even without outright supernatural intervention, there seems to be some 'flexibility' built in -- Hobbits are essentially Men, and they live longer: the Old Took made it to 150. So the Gift of Men doesn't seem to demand an absolute upper limit of circa 120 as we see in real life.
And the Valar can't undo the gifts of Ilúvatar, but they could make the Númenóreans live to 200-plus.
The Ringwraiths (and Gollum) are much older than that, though. And the life-prolonging effect of the Rings seems to be somewhat different. Aragorn doesn't seem to have lost anything by his extended lifespan, but the Nazgûl have literally faded away, their bodies seem to have lost most of their 'solidity' - Gandalf says
"[…] the black robes are real robes that they wear to give shape to their nothingness when they have dealings with the living."
Gollum is pretty twisted by the Ring, and even Bilbo (who had the Ring for far less time) describes the effect as:
"[…] sort of stretched, if you know what I mean: like butter that has been scraped over too much bread."
It seems the Rings 'cheat' by increasing 'quantity' of life without increasing 'quality'.
I think that there is an ambiguity here that is at the root of an answer to this question. From the Silmarillion:
For it was not permitted to the Valar to withhold Death from him [Beren], which is the gift of Ilúvatar to Men.
The Valar indeed may not withdraw the gift of death, which comes to Men from Ilúvatar, but in the matter of the Half-elven Ilúvatar gave to them the judgement; and they judged that to the sons of Eärendil should be given choice of their own destiny.
I don't think that Tolkien is ever more explicit than this and ever says that Men cannot cheat death, only that it is not permitted. It is forbidden by Eru. For the Valar to grant greatly extended life to a Man would be an act of rebellion on their part and a great evil, and for that reason they freely choose not do it without express permission of Eru.
On the other hand, we see beings already in rebellion -- Morgoth and Sauron -- doing just that: Húrin is bound to a chair on the face of Thangorodrim unable to die and Sauron's Rings of Power confer a sort of living death on Men who wear them.
Mortals, too, can condemn themselves to undeath by great evil: I suggest that the Dead of Dunharrow condemned themselves by reneging on their oath. Isildur could have freed them by releasing them from their oath -- any person can release another from an obligation to themselves -- but he didn't. That required no special magical powers, merely his legal and moral right.
Bottom line: Unending, resurrected, or extended life for mortals is either (1) the gift of Eru (usually by granting the Valar permission to act) or (2) the consequence of a great evil on the person's own part or (3) because of rebellion of one of the Ainur against Eru. Extension of a mortal's life is definitely possible, but it always is an act of Eru or involved an act of rebellion against Eru -- which Eru permits because of the freedom he granted his creations.