One that the PCs can gain I mean. I would think it would be a mythic feat or mythic path ability as even fast healing 1 would be pretty powerful for PCs, as even if they are almost dead they can just wait a few minutes out of combat and be good as new, whereas monsters and NPCs are usually only around for one fight so it doesn't matter.
I mean, you can get Fast Healing 1 for 90k. So it is something players can get without mythic. Taking normal wealth by level and not being allowed to spend more than 50% of wealth on a single item, a level 14 PC could start with this.
Make a ring of Infernal Healing...
There are bardic performances (even a bardic performance spell) that grant fast healing and it's possible to make a bardic performance last forever. While not technically permanent it's pretty close.
Similarly, a white mage arcanist can grant fast healing to allies using their arcane pool and one of their exploits lets them refill said arcane pool indefinitely. So again while not technically permanent it is close.
Certain rage abilities grant fast healing while you rage, though I can't think of anything atm that would allow a character unlimited rage (a level 20 capstone ability maybe?).
A life shaman's spirit animal gains fast healing. There may be a way though spells or abilities for the character to benefit from it.
Weirdly, I did find a mythic ability that grants fast healing but it's limited to a short duration and you have to spend a mythic point to trigger it, making it weaker than some of the things mentioned above.
Of course if it's ok to get it through a magic item there are Boots of the Earth. That give you fast healing at will, you just can't run around while it's active.
At level 11, the synthesist summoner could have fast healing as long as he's awake. Or the regular synthesist could have it at all the time at level 18 with greater aspect.
At level 12, the goliath druid could have regeneration 5 for basically the whole day, which is a lot better than fast healing.
Get bitten by a Vampire and gain the Vampire template.
Goliath Druid seems like the best, though if you want to be another class I would say boots of the earth, although I am wondering if you can turn them on while lying down as long as your feet are still touching the earth.
I proposed a new type of magic item in the homebrew section not too long ago. It was for a fast healing effect. I set the price at fast healing rate ^ 2 * 2000 gp.
If you are supposed to get magic protection using some formula instead of permanent mage armor spell, then why couldn't I do the same with fast healing? Make it work in my favor. Get a ring of fast healing 1 for the price of 2000 gp, instead of saving up 90k for an expensive ring of regeneration.
I think its a good deal. The alternative is you could buy 3 wands of cure light wounds for 2250 gp.
If you take out the regeneration effect, a Ring of +1 Fast Healing is still somewhere between 40-50k. A ring of fast healing 1 restores 10 HP in a minute, 600 in an hour, and 14k in a day. That is not something a level 4 character should be able to afford. 3 Wants of Cure Light Wounds heal a maximum of 1350, but more likely ~800 HP.