We present theoretical and experimental results on the mechanical damping of an atomic force microscope cantilever strongly coupled to a self-assembled InAs quantum dot. When the cantilever oscillation amplitude is large, its motion dominates the charge dynamics of the dot which in turn leads to nonlinear, amplitude-dependent damping of the cantilever. We observe highly asymmetric line shapes of Coulomb blockade peaks in the damping that reflect the degeneracy of energy levels on the dot. Furthermore, we predict that excited state spectroscopy is possible by studying the damping versus oscillation amplitude, in analogy with varying the amplitude of an ac gate voltage.