A statement on the League Two club's official website confirmed the news, citing a desire from Clarke to spend more time with his family after recently becoming a father.
The statement read: "Following the recent birth of his baby girl and developments in his professional career, Matt feels he can no longer commit the required time to fulfil the role to its maximum.
"The club would like to thank Matt for his efforts during his time as a director and look forward to seeing him back at Adams Park in the near future."
Clarke told Bucks Free Press: "I wish the club every success. I'm still a massive, massive supporter and they have a great team on and off the pitch.
"It was a really tough decision but it will be the right one for me.
"The club have been brilliant about it, which I very much appreciate."