As far as I can tell, All Time Lords have the ability to generate.
It's part of their 2-heart DNA.
The Master, The Doctor's Daughter (cloned from the doctor), and Rasalon all were able to change form and regenerate.
Time Lords, according to the new cannon exists in a space bubble on Gallifrey 's Floor.
So I suppose other creatures could exist outside the bubble.
See the image below.
Yes other creatures live on Gallifrey and not all of them are Time Lord:
Native animals of Gallifrey include flutterwings, trunkikes, yaddlefish (Blind Fury), flubbles, tafelshrews, plumboles (The Ghosts of N-Space), rovies, mice, cats and, of course, the Gallifreyans themselves.
In the past, the dinosaur-like Gargantosaurs lived on the planet.
Oddly enough, before its complete annihilation from time and space, no animal had gone extinct from the planet.
See the Gallifrey The Time Lords originated from the Gallifreyan species.
The difference is noted that "Time Lords are possibly just Gallifreyans in possession of a TARDIS."
on that wiki site.
I suspect just a change of name and some features.
Whether regeneration is a feature only to the time lords, I speculate that it is not.
Artificially, a human tried it in "Lazarus" (Season 3 or 4, I don't remember).
The people outside could either be "banished" aliens and Time Lords, or older Gallifreyans.
Some Helpful places... Doctor
Who (Main Wikipedia) TimeLord from the TARDIS Index File
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I think she is talking about the Invasion Of Time where Leela is chucked out of the Capitol and meets up with the "outsiders".
This doesn't really answer the question as I believe the question has never been asked in the series.
Sounds like a good proposition for a story though.
My suspicion is that regeneration is something that the Time Lords developed themselves as a technology.
I can't imagine how such a thing could evolve naturally.
So in that case the answer would be no.
In the Invasion of Time I think it is suggested that the "outsiders" are people who were previously Time Lords but had rejected that lifestyle and chose to live outside the Capitol.
So they possibly could still regenerate.
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In Underworld it is told how the Time Lords had developed technology to replicate regeneration in other species and given that technology to the Minyans.
It was explained that the Time Lords developed technological techniques to replicate their biological abilities.
The disastreous results of this resulted in the Time Lord adoption of the principle of (almost) non-interference.
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The new series has now established that the ability to regenerate has nothing to do with Gallifreyan origin, since River has that ability.
Whether or not she counts as a Time Lady has not yet been established, but her regeneration is explicitly stated as being due to the fact that ... (spoilers, sweetie) ...
she was conceived in the Tardis.
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In "A Good Man Goes To War", the 11th Doctor remarks that Time Lords evolved the ability to regenerate over billions of years, because there was a temporal schism on their own planet.
This genetic alteration due to evolving next to unfiltered time implies that it's a species wide ability unrelated to their binary vascular system, which other species also have, presumably without regeneration.
(Such as the species in "The Girl Who Waited")
As for Susan Foreman - bear in mind that at that time, regeneration didn't even exist in the series, so don't expect too much first or second season consistency with later canon .
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In the Invasion of Time, the outsiders leader, Nesben tells Leela that they "are Time Lords, or rather we were until we decided to drop out.
All that peace and eternal tranquillity, we decided to get back to nature out here."
(Outside of the Citadel)
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Well see I think it has to do with the tardis, because of river song.
She wasn't cloned from the doctor or born on galifrey.
She was conceived in the tardis.
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I saw in an question that Susan Foreman might not be a Time Lady. This made me think if she would regenerate like the Doctor. Are all people from Gallifrey Time Lords? I remember an episode where there were people who seemed to live outside of the Time Lord world. Do they all regenerate?
Are all Gallifreyans able to regenerate?
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<doctor-who>
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Regeneration may be something that the Time Lords developed themselves. In "A Good Man Goes To War", the 11th Doctor remarks that Time Lords evolved the ability to regenerate over billions of years, because there was a temporal schism on their own planet.   However, some speculate that regeneration is not an exclusive feature to time Lords. Indeed, the new series has now established that the ability to regenerate has nothing to do with Gallifreyan origin, since River has that ability. That said, some maintain that all  Time Lords  have the ability to generate, as The Master, The Doctor's Daughter (cloned from the doctor), and Rasalon all were able to change form and regenerate. All Time Lords have the ability to regenerate and they probably developed the technology to do so. However, regeneration is not limited to Time Lords. With regard to Susan Foreman, she appeared in the series before regeneration existed.
All Time Lords have the ability to regenerate. Regeneration is not specific to Time Lords, although they developed the technology for it. Regeneration has nothing to do with Galiifreyan origin.  When Susan Foreman was in the series, regenration did not exist.