  We present a systematic calculation of the cross section for the
lepton-proton bremsstrahlung process l + p --> l' + p + gamma in chiral
perturbation theory at next-to-leading order. This process corresponds to an
undetected background signal for the proposed MUSE experiment at PSI. MUSE is
designed to measure elastic scattering of low-energy electrons and muons off a
proton target in order to extract a precise value of the proton's r.m.s.
radius. We show that the commonly used peaking approximation, which is used to
evaluate the radiative tail for the elastic cross section, is not applicable
for muon-proton scattering at the low-energy MUSE kinematics. Furthermore, we
point out a certain pathology with the standard chiral power counting scheme
associated with electron scattering, whereby the next-to-next-to-leading order
contribution from the pion loop diagrams is kinematically enhanced and
numerically of the same magnitude as the next-to-leading order corrections. We
correct a misprint in a commonly cited review article.
