Now 35, Mee leaves the Bees with his contract running down, having joined from Burnley in 2022.
Mee told the club's website: "This is a unique club full of fantastic people, and I am truly proud and grateful to have had the opportunity to play here.
"I still have a lot left to give on the pitch, and will definitely try to do it somewhere else next season.
"Now I hope we can have a strong end to the season."
Bees manager Thomas Frank added: “Ben has been fantastic for us. His first season was outstanding; he was a big part of us finishing ninth in the Premier League.
“He has not had as many minutes in the two seasons since then, but he has always performed when called upon and been invaluable off the pitch.
"He has helped us build a strong culture and shared his knowledge with the younger players about how to be a top defender.
“Ben is a top pro and a top person - he will be missed. We wish him all the best for the future.”