Quantum light–matter interactions of bound electron systems have been studied extensively. By contrast, quantum interactions of free electrons with light have only become accessible in recent years, following the discovery of photon-induced near-field electron microscopy (PINEM). So far, the fundamental free electron–light interaction in all PINEM experiments has remained weak due to its localized near-field nature, which imposes an energy–momentum mismatch between electrons and light. Here, we demonstrate a strong interaction between free-electron waves and light waves, resulting from precise energy–momentum phase-matching with the extended propagating light field. By exchanging hundreds of photons with the field, each electron simultaneously accelerates and decelerates in a coherent manner. Consequently, each electron’s quantum wavefunction evolves into a quantized energy comb, spanning a bandwidth of over 1,700 eV, requiring us to extend the PINEM theory. Our observation of coherent electron phase-matching with a propagating wave is a type of inverse-Cherenkov interaction that occurs with a quantum electron wavefunction, demonstrating how the extended nature of the electron wavefunction can alter stimulated electron–light interactions.Energy–momentum phase-matching enables strong interactions between free electrons and light waves. As a result, the wavefunction of the electron exhibits a comb structure, which was observed using photon-induced near-field electron microscopy.