We classify singular Enriques surfaces in characteristic two supporting a rank nine configuration of smooth rational curves. They come in one-dimensional families defined over the prime field, paralleling the situation in other characteristics, but featuring novel aspects. Contracting the given rational curves, one can derive algebraic surfaces with isolated ADE-singularities and trivial canonical bundle whose Q_l-cohomology equals that of a projective plane. Similar existence results are developed for classical Enriques surfaces. We also work out an application to integral models of Enriques surfaces (and K3 surfaces).