1) The Irmin Git store, which is designed to run both on Xen unikernels and in JavaScript. Thomas Leonard is working on integrating the latter, with very promising results for a React-style interface in the browser. Thomas Gazagnaire is integrating the Xen backend at the moment. Expect an announcement of both of these just as soon as they're ready.
2) The Rump Kernel project has very promising results with MySQL running as a unikernel, so we're working on integrating a MirageOS frontend with a MySQL backend unikernel.
3) The Ctypes library (0.4) that's just been released supports Xen compilation for OCaml bindings, so writing bindings to Sqlite would let you link in directly with a unikernel; https://github.com/ocamllabs/ocaml-ctypes . There's an ORM we wrote ages ago with this in mind: https://github.com/mirage/orm
So the db efforts are full steam ahead at the moment, but no fully released one yet for production. Keep an eye on the blog at http://openmirage.org, or drop a note to <mirageos-devel@lists.xenproject.org> for your intended usecase.
One interesting advanced prototype that Dave Scott from Citrix built is the XenStore database as a unikernel (that powers inter-VM communication) on top of Irmin. Video: https://www.youtube.com/watch?v=DSzvFwIVm5s and a presentation at the Xen Developers Summit: http://decks.openmirage.org/xendevsummit14#/