  We explore hypothetical vector particles, dark photons $\gamma'$, which mix
with the Standard Model photons and thus mediate interactions with charged
particles into the hidden sector. We study the elastic proton bremsstrahlung of
dark photons with masses 0.4-1.8 GeV, relevant for direct searches with proton
accelerators. A key feature of our calculation is that it explicitly considers
the non-zero momentum transfer between protons in the process $pp\rightarrow
pp\gamma'$. We compare the obtained differential and full bremsstrahlung cross
sections with the results of other authors. Our calculation agrees well (up to
3-9 % corrections) with the Weizsacker-Williams approximation that confirms its
applicability for proton beams. Then we refine predictions for the dark photon
production with proton beams of energy 30 GeV, 70 GeV, 120 GeV and 400 GeV
relevant for past, present and future experiments considered in literature.
