As far as I know, the answer is “no”.
There was an experimental IMAP server a long time ago with an Irmin backend, but as far as I know, it was never more than an experiment. It was part of the Mirage constellation of projects.
A quick internet search yields GitHub - gregtatcam/imaplet-lwt: imaplet revamp with Lwt, and indeed it looks pretty much abandoned 
Cheers,
Nicolas