We derive an intrinsic contribution to the nonadiabatic spin torque for nonuniform magnetic textures. It differs from previously considered contributions in several ways and can be the dominant contribution in some models. It does not depend on the change in occupation of the electron states due to the current flow but rather is due to the perturbation of the electronic states when an electric field is applied. Therefore it should be viewed as electric-field-induced rather than current-induced. Unlike previously reported nonadiabatic spin torques, it does not originate from extrinsic relaxation mechanisms or spin-orbit coupling. This intrinsic nonadiabatic spin torque is related by a chiral connection to the intrinsic spin-orbit torque that has been calculated from the Berry phase for Rashba systems.