The 23-year-old's previous deal had expired earlier this week, but new terms were offered to keep him at Old Trafford.
Mee has played regularly for Manchester United's Under-21s side, making 27 appearances for the academy. His experience at senior level, however, has been limited.
He made four appearances for Altrincham in the National League during the 2022-23 campaign.
Mee has also had spells with Trafford, Runcorn Linnets and Witton Albion.
Despite his limited first-team minutes, he is regarded as a valuable member of the squad and features regularly in training.
Announcing the contract extension, Manchester United said: "The academy graduate will continue to provide excellent support and competition for the goalkeeper group.
"The Birmingham-born stopper, who has represented Northern Ireland at youth level, has played a key role in first-team training in recent years, whilst representing United's Under-21s side in competitive fixtures."
"The impressive 23-year-old is continuing to develop with this latest contract renewal a further reward for his contributions. Everyone at the club would like to congratulate Dermot on his new deal," the statement added.
Fellow goalkeeper Andre Onana is leaving Manchester United for another loan spell at Trabzonspor.
Despite this, Mee is currently expected to be only the club's fourth-choice goalkeeper next season.
Senne Lammens established himself as United's number one in September after Onana's departure for Trabzonspor. Altay Bayindir was briefly first choice under Ruben Amorim before Lammens was introduced for the win over Sunderland, and has since become back-up to the Belgian.
Veteran goalkeeper Tom Heaton remains at the club after signing a new one-year contract at the start of June.
The 40-year-old is seen as one of the leaders in the dressing room, despite making just three senior appearances since returning to United in 2021.