We report on a strong nonlinear behavior of the photogalvanics and photoconductivity under excitation of HgTe quantum wells (QWs) by intense terahertz (THz) radiation. The increasing radiation intensity causes an inversion of the sign of the photocurrent and transition to its superlinear dependence on the intensity. The photoconductivity also shows a superlinear raise with the intensity. We show that the observed photoresponse nonlinearities are caused by the band-to-band light impact ionization under conditions of a photon energy less than the forbidden gap. The signature of this kind of impact ionization is that the angular radiation frequency omega = 2 pi f is much higher than the reciprocal momentum relaxation time. Thus the impact ionization takes place solely because of collisions in the presence of a high-frequency electric field. The effect has been measured on narrow HgTe/CdTe QWs of 5.7 nm width; the nonlinearity is detected for linearly and circularly polarized THz radiation with different frequencies ranging from f = 0.6 to 1.07 THz and intensities up to hundreds of kW/cm(2). We demonstrate that the probability of the impact ionization is proportional to the exponential function, exp(-E-0(2)/E-2), of the radiation electric field amplitude E and the characteristic field parameter E-0. The effect is observable in a wide temperature range from 4.2 to 90 K, with the characteristic field increasing with rising temperature.