Immortality

Immortality was the ability to live forever, or at least a vastly long time. Most beings called immortal were not also indestructible, but there were exceptions.

Attempts at gaining immortality
Lord President Borusa determined that Rassilon had discovered the secret of true immortality, and used the first five incarnations of the Doctor to obtain it. This turned out to be the immortality of a living statue. (TV: The Five Doctors)

According to Iris Wildthyme, she defeated Morbius's attempt to obtain Rassilon's gift of immortality. (PROSE: The Scarlet Empress)

Queen Xanxia of Zanak attempted to create an immortal body for herself using the energies stolen from other planets. (TV: The Pirate Planet)

Mawdryn and his group of scientists stole a metamorphic symbiosis regenerator from the Time Lords and used it on themselves to gain immortality, only to find that theirs was one of unending mutation and regeneration. (TV: Mawdryn Undead)

Astrolabus, who was in his final incarnation, attempted to gain immortality by stealing Voyager's star charts. (COMIC: Once Upon a Time Lord)

Qataka, afraid of death and having heard stories of the Time Lords, experimented with cybernetics in an attempt to gain immortality. (PROSE: Timewyrm: Genesys)

Professor Richard Lazarus invented a device which would reverse the ageing process, providing a form of immortality, but it also mutated him into a monster. (TV: The Lazarus Experiment)

The Family of Blood pursued the Tenth Doctor in an attempt to gain his body and become immortal. As punishment for their crimes, the Doctor granted them immortality in various unpleasant ways. (TV: The Family of Blood)

Joshua Naismith attempted to use the Vinvocci Immortality Gate to make his daughter immortal. (TV: The End of Time)

Angelo Colasanto attempted to make himself immortal after seeing Jack Harkness die and come back to life. (TV: End of the Road)

When the Great Intelligence were uploading humans' souls to the Wi-Fi in 2013, Miss Kizlet described it as "immortality, only fatal." (TV: The Bells of Saint John)

Gained immortality
According to Greek mythology, one of the Greek goddesses asked Zeus to make Tithonus immortal, but she forgot to ask for eternal youth. He became so ancient and decrepit, he eventually shrank into a cicada. He begged for death. (TV: Rendition)

In medieval times, a Viking girl named Ashildr was granted immortality when, after her death, a Mire restoration chip was placed in her head. This device would keep repairing her, so she would never die. The Twelfth Doctor later gave Ashildr another chip to give to someone she would want to keep with her. (TV: The Girl Who Died) Her immortality remained even by the 21st century, 1300 years after her initial death, and ultimately all the way up to the last 5 minutes of the universe's ultimate end, outliving every other immortal in Doctor Who. (TV:Hell Bent (TV story)) The second chip was subsequently used to revive highwayman Sam Swift, though the circumstances left the Doctor uncertain as to whether the result would be immortality in his case. (TV: The Woman Who Lived)

The Cybermen gained immortality by replacing their body parts with cybernetic ones, but they lost their emotions and humanity in the process. (TV: The Tenth Planet)

The race known as the Olympians had the power of immortality. (COMIC: The Life Bringer)

Jack Harkness, after being resurrected by the Bad Wolf, would always come back to life after being killed, and his ageing was slowed down. The Tenth Doctor explained that when Rose Tyler was the Bad Wolf she brought him back but couldn't control the power and brought him back forever. (TV: The Parting of the Ways, Everything Changes, Utopia) He lost this immortality on Miracle Day, (TV: The New World) but regained it when the Miracle was ended. (TV: The Blood Line)

Suzie Costello, after being brought back to life, was unable to die again until the resurrection gauntlet that was keeping her alive was destroyed. (TV: They Keep Killing Suzie)

Ursula Blake gained immortality when she became a face on a concrete slab. (TV: Love & Monsters)

Katsura Sato gained immortality after having been rescued by Nanodrones (COMIC: The Road to Hell), but lost it when Kroton took control of the Glory. (COMIC: The Glorious Dead)

Thomas Vaughan discovered the secret of immortality in the mid-17th century. (AUDIO: Hidden)

In an alternate timeline, the Nimon granted immortality to Sebastian Grayle. (AUDIO: Seasons of Fear)

In another alternate timeline, Wolfgang Amadeus Mozart was given immortality in 1791 by one of his clones from the far future. He later came to regret this as this meant that he lived long enough to become a hack. (AUDIO: My Own Private Wolfgang)

In another alternate timeline, the inhabitants of Stockbridge were given immortality between 1950 and 2009, courtesy of Viridios. Towards the end, they begged for death. They were given it when the timeline was reversed and they died when they were originally intended to. (AUDIO: The Eternal Summer)

A tarot card reader must have gained some form of immortality, or at least reduced ageing, as, between the years 1897 and 2008, despite a gap of more than 110 years, her appearance remained unchanged. (TV: Fragments, Dead Man Walking)

Grayvorn gained immortality from the Relic. (AUDIO: Excelis Rising)

Erasmus Darkening gained immortality. (TV: The Eternity Trap)

In 2011, during an event dubbed by the world press as "Miracle Day", the human race stopped dying due to Jack Harkness' immortal blood being introduced to The Blessing. It was reversed when Jack's mortal blood was introduced. (TV: The New World, The Blood Line)

Rex Matheson apparently gained immortality through means not understood after a complete transfusion of Captain Jack's then-mortal blood during the Miracle Day events. (TV: The Blood Line)

The disembodied heads of the members of the Order of the Headless lived on, even when they were reduced to skulls. (TV: The Wedding of River Song)

Shara created an energy converter which enabled him to transfer his life energy into a time loop in the Time Vortex, giving him eternal bliss. Jal Dor Kal lived for centuries by absorbing energy from Shara's relic. (AUDIO: Nekromanteia)

Vilus Krull used the power of the Dark Flame to transfer his mind into different host bodies upon their deaths, allowing him to live for centuries. (AUDIO: The Dark Flame)

The Mechanistrians gained immortality by having their brains placed in machines. (PROSE: Peril in Mechanistria)