Reticulum II (Ret II) is a satellite galaxy of the Milky Way and presents a prime target to investigate the nature of dark matter (DM) because of its high mass-to-light ratio. We evaluate a dedicated INTEGRAL observation campaign data set to obtain $gamma$-ray fluxes from Ret II and compare those with expectations from DM. Ret II is not detected in the $gamma$-ray band 25--8000 keV, and we derive a flux limit of $lesssim 10^{-8},mathrm{erg,cm^{-2},s^{-1}}$. The previously reported 511 keV line is not seen, and we find a flux limit of $lesssim 1.7 times 10^{-4},mathrm{ph,cm^{-2},s^{-1}}$. We construct spectral models for primordial black hole (PBH) evaporation and annihilation/decay of particle DM, and subsequent annihilation of positrons produced in these processes. We exclude that the totality of DM in Ret II is made of a monochromatic distribution of PBHs of masses $lesssim 8 times 10^{15},mathrm{g}$. Our limits on the velocity-averaged DM annihilation cross section into $e^+e^-$ are $langle sigma v rangle lesssim 5 times 10^{-28} left(m_{rm DM} / mathrm{MeV} right)^{2.5},mathrm{cm^3,s^{-1}}$. We conclude that analysing isolated targets in the MeV $gamma$-ray band can set strong bounds on DM properties without multi-year data sets of the entire Milky Way, and encourage follow-up observations of Ret II and other dwarf galaxies.
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