Permanently ban

No.

Only if the ban was a mistake.

Or if it was a permanent suspension due to the account being compromised, of course. But if we're talking behavioral bans, then yeah, only if a mistake was made. And they rarely are.

There are certain permanent bans that are just placeholders until you resolve the issue with Riot. Such as charge-backs and account suspected of being compromised.

If it was a behavioral ban (that wasn't issued in error), the chances are extremely unlikely. You can try through a support ticket, and you'll have present a pretty damn convincing argument that the behavior that led to the behavioral ban is a thing of the past.