  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).
