An ensemble of inelastically colliding grains driven by a horizontally vibrating wall in two dimensions exhibits clustering. Working in the limit of nearly elastic collisions and employing granular hydrodynamics, we predict, by a marginal stability analysis, a spontaneous symmetry breaking of the laterally uniform clustering state. Two-dimensional steady-state solutions found numerically describe laterally periodic clustering states. Well within the instability region the density of these states is strongly peaked, with most of the granulate located in "density islands." Time-dependent granular hydrodynamic simulations show that strongly peaked states can develop from small-amplitude single-mode density perturbations.