We report the first demonstration of high-harmonic generation from plasma mirrors at a 1 kHz repetition rate. Harmonics up to nineteenth order are generated at peak intensities close to 10¹⁸ W/cm² by focusing 1 mJ, 25 fs laser pulses down to 1.7 μm FWHM spot size without any prior wavefront correction onto a moving target. We minimize target surface motion with respect to the laser focus using online interferometry to ensure reproducible interaction conditions for every shot and record data at 1 kHz with unprecedented statistics. This allows us to unambiguously identify coherent wake emission as the main generation mechanism.