The phase diagram of massless quantum electrodynamics in three space-time dimensions as a function of fermion flavor number N exhibits two well-known phases: at large N>Ncconf the system is in a conformal gapless state, while for small N<NcχSB the fermions are expected to develop a dynamical mass due to spontaneous chiral symmetry breaking. Using ϵ expansion near the lower critical dimension of 2, as well as the recent results on the generalization of the F theorem to continuous dimension, we show that Ncconf>NcχSB. There is therefore an intermediate range of values of N at which a third phase is stabilized. We demonstrate that this phase is characterized by spontaneous breaking of Lorentz symmetry, in which a composite vector boson field acquires a vacuum expectation value with the fermions and the photon remaining massless.