The Denmark winger has signed a new deal to 2030.
“I am very happy that Mikkel has signed a new contract with us,” said Frank.
“This season, he is starting to show what we believed we’d get from him. We knew that wouldn’t happen straight away because of the time he was out before he came here, but it ended up being longer than expected - both for him and us.
“Mikkel was then unlucky with another injury but, after two years of relatively consistent training and a good environment - and with patience both ways - we are now seeing the fruit of the work.
“He has been one of our best players this season; we are very dangerous when he is playing.
“But there is more to come from Mikkel. It has been half a season, now he needs to perform for a full season. We need to add layers, but I am happy with what he is doing.”
Damsgaard also told the club's website: "I’ve enjoyed my years here so it was a no-brainer for me to extend - I’m very happy.
“This season has been a lot better for me personally with a lot more playing time, but I’ve always enjoyed the atmosphere at the club, time with my team-mates and the way things have been set up.
“I’m very proud and happy to be here. I’m very happy to see how the fans support me. It makes you give that extra five per cent and I’m very thankful for the fans we have.
“With the playing time now coming, it was a pretty easy decision to extend."