So Cambions while generally typed as "fiends" are actually of half fiend / half mortal-humanoid parentage.  Most fiends, I believe, but at least demons and devils are technically immortal aside from death on their own plane so I don't Lichdom would be something they're interested in let alone actually being capable of becoming.  By that I mean, the mortal soul aspect is probably what allowed Acerek to become a lich, and he does seem to be something as an exceptional character.  I don't think Devils would tolerate lichdom in its hierarchy less Orcus somehow get a foothold.  Demons I don't see going for it either because my guess it would stunt the ambition demon's desire to be a demon lord.  That said, I'm sure Orcus has a coterie of disciplines and minions whose grasp of necromancy and undeath probably rivals that of most Liches.  On the infernal side, maybe Levistus would be a similar tutor, not sure if any particular Arch Devil has domain over undead or necromancy or it the general "boss" of Bone Devils.

I agree that whether or not it's possible, the bigger question is why an already immortal being would want to became a lich? The purpose of lichdom being to be immortal.

That being said, fiends can and do (for some) store part of their "soul" in phylactary (not sure it's called the same but it works similarly). It allows fiends who have been killed outside their plane to quickly go back to some preserved level of power rather than start from scratch.

TL:DR at the end.

I know that in previous editions, lichfiends (literally, that was the name for them) was an option (3.5, libris mortis splatbook). So you are on the right track.

The problem as others have stated, is why. Fiends already have their own sources of immortality with their own strengths and weaknesses, most notably that they usually reconstitute themselves on their home plane if killed anywhere else. For one to bind themselves to a phylactery and reform at that point when killed doesn't necessarily make them stronger from a lore point-of-view, it just changes the conditions under which they operate. 

But lets go ahead and look at a scenario in which a change would be undergone, maybe for a devil. Devils are Lawful Evil, meaning they are only out to benefit themselves and their superiors, while playing within a strict hierarchy of power. An imp dreams of ascending to the status of a pit fiend, while the pit fiend clings desperately to the power that it has attained against all the forces trying to sabotage it.

So from a devil's point of view, perhaps one has gotten tired of this flavor of eternal torment and is craving another. Some curious cambion looks around, and carefully plans a way to fake their death to their superiors, then high-tails it to the prime material plane where they learn the requisite spells that will intrinsically change the way in which not only they will live forever, but in the way that the world reacts to their existence.

On a personal note, I love the lore idea that, for the more powerful outsiders like fiends, celestials, etc. names have power, simply being 'x' means you are at the mercy of 'y' powers even if you don't explicitly agree, etc.

So from that perspective, going from being a devil to being an undead frees you from all your inherent obligations and responsibilities to your previous masters in hell. You would still have to look over your shoulder for the next hit squad from your previous employers, but hey- what lich doesn't have to do that?

TL:DR - a fiend becoming a lich isn't as much of a big thing as it would be for us mortals, but it does change the way in which you can operate in some dnd-universes.