We consider the nonlinear interaction problem of surface waves with a tethered near-surface buoy. Our objective is to investigate mechanisms for nonlinear short surface wave generation in this complete coupled wave–buoy–cable dynamical system. We develop an effective numerical simulation capability coupling an efficient and high-resolution high-order spectral method for the nonlinear wave–buoy interaction problem with a robust implicit finite-difference method for the cable–buoy dynamics. The numerical scheme accounts for nonlinear wave–wave and wave–body interactions up to an arbitrary high order in the wave steepness and is able to treat extreme motions of the cable including conditions of negative cable tension. Systematic simulations show that beyond a small threshold value of the incident wave amplitude, the buoy performs chaotic motions, characterized by the snapping of the cable. The root cause of the chaotic response is the interplay between the snapping of the cable and the generation of surface waves, which provides a source of strong (radiation) damping. As a result of this interaction, the chaotic buoy motion switches between two competing modes of snapping response: one with larger average peak amplitude and lower characteristic frequency, and the other with smaller amplitude and higher frequency. The generated high-harmonic/short surface waves are greatly amplified once the chaotic motion sets in. Analyses of the radiated wave spectra show significant energy at higher frequencies which is orders of magnitude larger than can be expected from nonlinear generation under regular motion.