We show that the van der Waals-like phase-separation instability of a driven granular gas at zero gravity, previously investigated in two-dimensional settings, persists in three dimensions. We consider a monodisperse granular gas driven by a thermal wall of a three-dimensional rectangular container at zero gravity. The basic steady state of this system, as described by granular hydrodynamic equations, involves a denser and colder layer of granulate located at the wall opposite to the driving wall. When the inelastic energy loss is sufficiently high, the driven granular gas exhibits, in some range of average densities, negative compressibility in the directions parallel to the driving wall. When the lateral dimensions of the container are sufficiently large, the negative compressibility causes spontaneous symmetry breaking of the basic steady state and a phase separation instability. Event-driven molecular dynamics simulations confirm and complement our theoretical predictions.