The ability to live for extremely long times without death generally implies the ability to heal from most damage. Given a long enough time, even humans can remove most minor scars. Animals like Lizards, Spiders, and Starfish all regrow lost limbs.

The same goes for Psychological trauma, to some extent. Over time, people cope. Some people don't get PTSD from some experiences while others do, and some people cope faster than others, so it's very possible your hypothetical race could be less psychologically frail than modern humans.

Longer lives lead to greater ambitions

These races are no longer confined to short lifespans; imagine what Thomas Edison or Albert Einstein or Nicholas Tesla would have accomplished if they were still alive today, and had access to today's technology. If these races also have great thinkers, they will not aspire to "improve the light bulb" in their lifetime - they will land people on Mars or establish the first lunar colony.

Ambitions lead to discovery

These races will inevitably discover new things at a faster rate — the same smart people will have time to accomplish more, and they will accumulate.

Discoveries may maintain youth

If a race has more thinkers, and more discoveries occur every day, surely some people will aspire to fix the scarring, limb loss, etc. that occur. Prosthetics will be improved, medical science and surgery accuracy will grow tenfold, flesh will be 3D-printable on commercial scales, brain damage will be reversed, and death will be a distant memory.

My disagreements with "resilience solves this problem"

• Resilience does not account for mental trauma. No matter how emotionally resilient or "unlikely to develop PTSD" individuals are, longevity causes inevitability. Eventually, a scarring event will occur, eventually dementia will set in, and eventually mental state will break down. Despite our attempts to extend the human life span through diet and medicine, dementia always finds a way.
  • People won't "cope". We treat mental illness because it isn't something people usually get over - if people could "cope" with mental illness on their own it wouldn't be a problem to humans anyways.
• Resilience does not account for limbs lost. Scars fade, but limbs do not regrow; extending the human lifespan will not give us this ability - millions of years of evolution MAY give this ability.
• Tissue regeneration is not the same as cancer negation. Even if a humanoid species naturally evolves regenerative abilities, the ability to create new cells at will does not solve the problem of eventual genetic errors. In fact, creating new cells faster will amplify the problems if a mutation occurs.

Complementing on what Landon Powell said... there are 2 methods to heal a wound: either you regenerate the damaged tissue as it were, or you substitute the damaged tissue by fibrous tissue, which is extremely resilient and binds the borders of the wound tightly together. The first method doesn't produce a scar, the other does.

Once fibrosis is established it is irreversible or reversible only after a long time, because you will need to substitute fibrous tissue by normal tissue, which is more difficult than just substituting a gap with normal tissue in the first place.

Now, all you have to do is create a species that uses the first method to heal wounds, but not the scarring method.

This isn't difficult to do. By definition, a long lived species has great healing properties. If not, they would die of cancer very soon. Cancer is derived from mutations on the genetic code. Age promotes these mutations, because the DNA "decays". Also, the more you live, the more divisions your cells undergo and the probability of errors on DNA replication at each cycle acumulares. And the more time passes by, the more likely is your exposure to environmental carcinogens.

So, if your species is so long lived as to survive cancer for decades or centuries, be assured that that species is able to regenerate tissue in an impressive way.

One way you could do this would be to make that species have ultra-sensitive stem cells. As soon as a lesion occurs, those stem cells would divide and diferenciate in a way as to replicate the entire tissue. The damaged tissue could even issue circulating signals so as to recruit stem cells from other bodyparts through blood. This happens in our body, BTW.

As for psychological damage, you could apply the same logic as with physical damage.... being coping analogous to regeneration and neurosis / PTS being analogous to scarring. Just make that species so emotionally resilient as to cope with every psychological damage they may experience.

Edit: To clarify, I'm not saying that PTSD is equal to scarring. Sorry if I wasn't clear. What I meant to say was that there are two ways of healing physical wounds - a healthy way, through regeneration, and a non healthy way, through scarring. Analogously, there are two ways to heal psychological wounds - a healthy way, through coping, and a non healthy way, through PTSD. You could create a species with such biological and psychological physiologies in place as to always go through the healthy route of healing.

Perhaps because they can molt?

If this long-lived species molts every ~5 years, then it would be a perfect way of removing anything but extreme scarring. Perhaps severe cases might require multiple molts. The molting could even be a factor in their longevity.

Also, they can regrow limbs while in the molting stage. Cover up the scars by giving them exotic joint structures that mask the appearance of the scarring there. So, a limbless member of this species would just deal with it for a few years and then, right before their next molt, rip off the remainder of the limb they wanted to regenerate. Wait a few weeks and voila! Shiny new arm!

If all members of the species only had visible scarring if it had occurred within the last 5 to 10 years, then there really wouldn't be much scarring to see.

I can't speak too much for psychological traumas, but I see two key possibilities for dealing with those.

1. The species has incorporated the concept of molting into their culture and there is a strong mental/spiritual component to a molt. Thus, they have an extremely powerful marker every 5 years on which to focus their mental healing, not to mention that the few weeks of motionlessness would give them the opportunity to sort through their problems as they were rejuvenating.
2. They don't form personal connections the way humans do. (Humans seem incapable of strong interaction without building a connection, either of love or hate.) With no love, there is no heartbreak. With no enemies, there is merely opposition to remove. There would be less opportunity for mental and psychological distress.

I like most of answers there, but let's look at that from different point of view.

As Landon Powell has said,

The ability to live for extremely long times without death generally implies the ability to heal from most damage.

But alternatively it may imply having enough commons sense (and highly-developed social norms) to cope with or avoid most damage.

Each time a human gets sunburnt the risk of suffering from skin cancer grows a little bit. Solution? Don't get sunburnt. Don't go out on sunny days (with "beautiful pale skin" being a free bonus). Unlike "supernatural regenerative abilities", this explanation scales very well, possibly for thousands or even billions years of lifespan.

A race that doesn't deal with other people directly.

1. Video phones
2. Actual avatars that look feel, and for all purposes seem real.

The lifeform would have a full sensor net that would allow them to sense/feel everything to a point. All 5 senses being sent back and forth to the real person/alien. To avoid mental scaring any long term pain would not be sent back to the user.

Avatar 101 dies, 102 roles out and takes their place.

To them life would be very similar to us playing our video games.

Dealing with death. After 50,100, or what years the computer controlling the avatars could probably emulate the person to a point where a person avatar goes on in their place basically forever.

While I agree all above answers I want to add another angle.

Culture and history. Anthropology...

Look at our world, some countries, tribes or other kind of societies are more peaceful or wise than other even though we all have the same brain and body structure.

Thats because some cultures have different priorities in their everyday life, their aproach to educate their children is different. etc...

Maybe they didn't invent money sometime in their history therefore they are unfamiliar the concept of kapitalism and all the war it brings. Maybe they found a more peaceful way to divide limited resources without cousing war. Things never had to become a survival issue so they never needed to end other tribes existance. Therefore they became a peaceful wise society.

Or maybe its total opposite, they had the most biggest and ugliest wars in their history and now they learned their lesson not to do that. Like how todays German schools teaching Hitler part of the history to their students.