We measured the phase velocity of surface acoustic waves (SAW's) with a scanning acoustic force microscope (SAFM) and achieved a maximum lateral resolution of 19.9 nm. The phase measurement of high-frequency waves with a slowly responding SAFM cantilever was perfomed by frequency mixing at its non-linear force curve. For Au layers of different thicknesses the SAW dispersion was studied on a lateral scale of 200 nm and compared to calculated data.