Permanently ban
Is there any way to get unban from permanent ban?
Is there any way to get unban from permanent 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.
And how to prove that ban was mistake?
Well, was it? If you were flaming, arguing, being excessively negative, and/or being toxic, and you passed through the 10-game restriction, 25-game restriction, and two-week ban steps, then no mistake was made. Or if you used racial/homohobic slurs and/or said other terrible things, you may have even skipped some steps.
If that doesn't apply to you, and you truly did nothing wrong, submit a support ticket and see if Riot staff can help you.
Post your chat logs & let us see. Don't do edits, Riot will found you out if you do so.
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.
How to get a support ticket?
Support link at the top -> "Submit a Request"