  The cross section which addresses the spin-flip transitions of a proton
(antiproton) interacting with a polarized non-relativistic electron or positron
is calculated analytically. In the case of attraction, this cross section is
greatly enhanced for sufficiently small relative velocities as compared to the
result obtained in the Born approximation. However, it is still very small, so
that the beam polarization time turns out to be enormously large for the
parameters of $e^{\pm}$ beams available now. This practically rules out a use
of such beams to polarize stored antiprotons or protons.
