The spontaneous emergence of structures from initially homogenous systems belongs to the most striking topics in natural science. Systems driven into deeply nonlinear regimes are theoretically difficult to describe and can produce states that do not exist in equilibrium. We observe the emergence of a stable square lattice density modulation from an initially homogenous, two-dimensional, radially symmetric Bose-Einstein condensate when periodically driving the two-particle interaction. We show theoretically that this state can be understood as an attractive fixed point of coupled nonlinear amplitude equations, which result from phonon-phonon interactions. As a self-stabilized state characterized by spontaneously broken translational symmetry, our results establish a novel quantum material related to supersolids.