Shift current-a photocurrent induced by light irradiating noncentrosymmetric materials in the absence of any bias voltage or built-in electric field-is one of the mechanisms of the so-called bulk photovoltaic effect. It has been traditionally described as a nonlinear optical response of a periodic solid to continuous wave light using a perturbative formula, which is linear in the intensity of light and which involves Berry connection describing the shift in the center of mass position of the Wannier wave function associated with the transition between the valence and conduction bands of the solid. Since shift current is solely due to off-diagonal elements of the nonequilibrium density matrix that encode quantum correlations, its peculiar space-time dynamics in response to femtosecond light pulse employed locally can be expected. To study such response requires to analyze realistic two-terminal devices, instead of traditional periodic solids, for which we choose paradigmatic Rice-Mele model sandwiched between two metallic electrodes and apply to it time-dependent nonequilibrium Green function algorithms scaling linearly in the number of time steps and capable of treating nonperturbative effects in the amplitude of external time-dependent fields. This reveals novel features:superballistictransport, signified by time dependence of the displacement, similar to t(nu)with nu > 1, of the photoexcited charge carriers from the spot where the femtosecond light pulse is applied toward the electrodes; and photocurrent quadratic in light intensity at subgap frequencies of light due totwo-photon absorptionprocesses that were missed in previous perturbative analyses. Furthermore, frequency dependence of the DC component of the photocurrent reveals shift current as a realization of nonadiabatic quantum charge pumping enabled bybreaking of left-right symmetryof the device structure. This demonstrates that a much wider class of systems, than the usually considered polar noncentrosymmetric bulk materials, can be exploited to generate nonzero DC component of photocurrent in response to unpolarized light and optimize shift-current-based solar cells and optoelectronic devices.