Crucially important for application of type-II superconductor films is the stability of the vortex matter--magnetic flux lines penetrating the material. If some vortices get detached from pinning centres, the energy dissipated by their motion will facilitate further depinning, and may trigger a massive electromagnetic breakdown. Up to now, the time-resolved behaviour of these ultra-fast events was essentially unknown. We report numerical simulation results revealing the detailed dynamics during breakdown as within nanoseconds it develops branching structures in the electromagnetic fields and temperature, with striking resemblance of atmospheric lightning. During a dendritic avalanche the superconductor is locally heated above its critical temperature, while electrical fields rise to several kV/m as the front propagates at instant speeds near up to 100 km/s. The numerical approach provides an efficient framework for understanding the ultra-fast coupled non-local dynamics of electromagnetic fields and dissipation in superconductor films.