We study electron acceleration in a plasma wakefield under the influence of the radiation-reaction force caused by the transverse betatron oscillations of the electron in the wakefield. Both the classical and the strong quantum-electrodynamic (QED) limits of the radiation reaction are considered. For the constant accelerating force, we show that the amplitude of the oscillations of the QED parameter $\chi$ in the radiation-dominated regime reaches the equilibrium value determined only by the magnitude of the accelerating field, while the averaged over betatron oscillations radiation reaction force saturates at the value smaller than the accelerating force and thus is incapable of preventing infinite acceleration. We find the parameters of the electron bunch and the plasma accelerator for which reaching such a regime is possible. We also study effects of the dephasing and the corresponding change of accelerating force over the course of acceleration and conclude that the radiation-dominated regime is realized both in cases of single-stage acceleration with slow dephasing (usually corresponding to bunch-driven plasma accelerators) and multi-stage acceleration with fast dephasing (corresponding to the use of laser-driven accelerators).