Millimeter-wave (mm-wave) communications and radar receivers must be protected from high-power signals, which can damage their sensitive components. Many of these systems arguably can be protected by using photonic limiting techniques, in addition to electronic limiting circuits in receiver front-ends. Here we demonstrate, experimentally and numerically, a free-space, reflective mm-wave limiter based on a multilayer structure involving a nanolayer of vanadium dioxide VO2, which experiences a heat-related insulator-to-metal phase transition. The multilayer acts as a variable reflector, controlled by the incident wave intensity. At low intensities VO2 remains dielectric, and the multilayer exhibits strong resonant transmittance. When the incident intensity exceeds a threshold level, the emerging metallic phase renders the multilayer highly reflective while safely dissipating a small portion of the input power, without damage to the limiter. In the case of a Gaussian beam, the limiter has a nearly constant output above the limiting threshold input.