Soft layered systems buckling to form surface patterns has been widely studied under quasistatic loading. Here, we study the dynamic formation of wrinkles in a stiff-film-on-viscoelastic-substrate system as a function of impact velocity. We observe a spatiotemporally varying range of wavelengths, which display impactor velocity dependence and exceed the range exhibited under quasistatic loading. Simulations suggest the importance of both inertial and viscoelastic effects. Film damage is also examined, and we find that it can tailor dynamic buckling behavior. We expect our work to have applications to soft elastoelectronic and optic systems and open routes for nanofabrication.