  We estimate the beam-normal single-spin asymmetry in elastic lepton-proton
scattering without employing the ultrarelativistic approximation. Our
calculation is relevant for analyses of muon scattering at energies of few
hundred MeV and below -- when effects of the muon mass become essential. At
such energies, the transverse polarization of the muon beam is expected to
contribute significantly to the systematic uncertainty of precision
measurements of elastic muon-proton scattering. We evaluate such systematics
using an example of the MUSE experiment at PSI. The muon asymmetry is estimated
at about 0.1\% in kinematics of MUSE and it is the largest for scattering into
a backward hemisphere.
