Several answers mention either outright ignoring the roll and doing whatever you want with the story, or an idea that magic should always trump skills, or else you should let the player shine at what they clearly want to play.
There's a few issues with these approaches.
Story taking precedence over mechanics is a good idea in theory, except it can easily feel like steamrolling the player.
If you make a character with an extremely high AC, and suddenly all the monsters are getting 30+ on their attack rolls, it feels like the world is specifically out to get you. If you roll a 40+ on a Stealth check and the town guard sees you anyway, you're going to feel rightfully feel cheated.
Likewise, if such insanely good skill rolls are always one-upped by magic, then it begins to feel like a game of The Almighty Wizard and His Amazing Friends, rather than Dungeons and Dragons (something D&D already arguably struggles with).
A specialized 20th level Rogue getting a 30+ on a disguise check being immediately detected without a save by a 3rd level Wizard casting a Detect Thoughts spell is bad enough, also stating that simple spells like Disguise Self should always beat that 30+ roll just because its "magic" is just adding salt to the wound.
Being a fan of the PC's and letting characters shine at what they are built for is fun, but it can become too much if it begins to outshine the other PC's.
A DM should be a fan of all the PC's, and they should each have a chance to shine, without one guy zooming in like Superman to fix all their problems, all the time.
This is where it sounds like you are at: the PC in question has a super high Persuasion score, and is using it to overcome an overabundance of challenges and it's taking away from the story and from other characters in the party.
To fix this issue, we ought to first identify more fully what the issue actually is. Is the issue that the player is abusing the skill? Should Persuasion not actually be able to do this? Is there such a thing as "impossible" for a Persuasion check? Someone has mentioned that a person is not going to give up their savings account to a stranger, no matter how good their Persuasion check is. Except, ironically, people do exactly that in real life. Gullible people are taken advantage of by con men all the time.
In game terms, "gullible" means having a low Wisdom and/or being untrained in Insight. A con man, on the other hand, is likely to have very high Charisma, proficiency in Deception, and possibly Expertise as well. So if we imagine that persuading someone to give up their savings account would be a Deception check, with perhaps advantage or disadvantage coming into it based on how well the con man is using personality traits of the mark (like being greedy, for a Nigerian Prince scam, or being compassionate, for a "help my dying grandma" sort of scam), and then the opposed roll is Insight to see through the lies, you could also have bonuses or penalties based on the stakes.
Convincing someone to give you their life savings might incur a huge penalty, because of what is at stake, but someone with 6 Wisdom and no Insight might still fall for it, given a persuasive con man (maybe beating the opposed check by 10 or more, for example).
So if the issue isn't whether Persuasion should be able to convince NPC's to do all this stuff (just increase the DC), what is it?
I would think the issue is A) whether the player actually can hit the target DC (once adjusted for mitigating factors), and B) how many opportunities are you giving the players to solve their problems through Persuasion.
For the second point, imagine instead you had a player who specialized in dealing massive amounts of physical damage. Maybe it's a barbarian, using a great axe. If 9/10 encounters are straight-forward ogre battles, with tactics like "I run up and hit it" being sufficient to pass, the barbarian is going to look like a god. Mix in some ghosts, flying enemies, spellcasters, social skill encounters, hidden enemies, etc. Now that barbarian shines when he gets the chance to engage in melee with a enemy, but the rest of the party can still feel good about their own contributions for the rest of the time.
For your Persuasion specialized character, maybe mix in some more dungeon crawling, exploration-based encounters, or simply NPC's who can't speak his language, but other party members can. That way the silver tongued devil still appropriately demonstrates his godlike charm when it applies, but hopefully won't disproportionately hog the spotlight for the whole session.