  We consider an electron-hole bilayer in the limit of extreme density
imbalance, where we have a single particle in one layer interacting
attractively with a Fermi liquid in the other parallel layer. Using an
appropriate variational wave function for the dressed exciton, we provide
strong evidence for the existence of the Fulde-Ferrell-Larkin-Ovchinnikov
(FFLO) phase in electron-hole bilayers with a large density imbalance.
Furthermore, within this unusual limit of FFLO, we find that a dilute gas of
minority particles forms excitons that condense into a two-dimensional
"supersolid".
