  We study the single-inclusive production of prompt photons in electron proton
collisions, $ep \to \gamma X $, for kinematics relevant at the Electron-Ion
Collider (EIC). We perform a perturbative calculation of the differential cross
section to next-to-leading order in QCD and to lowest order in QED. We consider
unpolarized collisions as well as scattering of longitudinally polarized
incident electrons and protons. We show that the cross sections are sensitive
to the parton distribution functions of photons inside the proton, which we
find to generate the dominant contributions in certain kinematical regions at
the EIC. We also investigate the effects of photon isolation on the unpolarized
and polarized cross sections.
