  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.
