With the advent of pure-spin-current sources, spin-based electronic (spintronic) devices no longer require electrical charge transfer, opening new possibilities for both conducting and insulating spintronic systems(1-9). Pure spin currents have been used to suppress noise caused by thermal fluctuations in magnetic nanodevices, amplify propagating magnetization waves, and to reduce the dynamic damping in magnetic films(10-14). However, generation of coherent auto-oscillations by pure spin currents has not been achieved so far. Here we demonstrate the generation of single-mode coherent auto-oscillations in a device that combines local injection of a pure spin current with enhanced spin-wave radiation losses. Counterintuitively, radiation losses enable excitation of auto-oscillation, suppressing the nonlinear processes that prevent auto-oscillation by redistributing the energy between different modes(12,15). Our devices exhibit auto-oscillations at moderate current densities, at a microwave frequency tunable over a wide range. These findings suggest a new route for the implementation of nanoscale microwave sources for next-generation integrated electronics.