We report the first demonstration of high-harmonic generation from plasma mirrors at a $1\mathrm{kHz}$ repetition rate. Harmonics up to nineteenth order are generated at peak intensities close to ${10}^{18}\mathrm{W}/{\mathrm{cm}}^2$ by focusing $1\mathrm{mJ}$ , $25\mathrm{fs}$ laser pulses down to $1.7\mathrm{\mu m}$ 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 $1\mathrm{kHz}$ with unprecedented statistics. This allows us to unambiguously identify coherent wake emission as the main generation mechanism.