We investigate single-cycle terahertz (THz) field-induced nonlinear absorption in doped silicon carbide. We find that the nonlinear response is ultrafast, and we observe up to 20% reduction of transmission of single THz pulses at peak field strengths of 280 kV/cm. We model the field and temperature dependence of the nonlinear response by a finite-difference time-domain simulation that incorporates the temporally nonlocal nonlinear conductivity of the silicon carbide. Nonlinear two-dimensional THz spectroscopy reveals that the nonlinear absorption has an ultrafast subpicosecond recovery time, with contributions from both sum-frequency generation and four-wave mixing, in the form of a photon-echo signal. The ultrafast nonlinearity with its equally fast recovery time makes silicon carbide an interesting candidate material for extremely fast nonlinear THz modulators.