  Minimal Dark Matter (MDM) stands as one of the simplest dark matter
scenarios. In MDM models, annihilation and co-annihilation processes among the
members of the MDM multiplet are usually very efficient, pushing the dark
matter mass above $\mathcal{O}(10)$ TeV in order to reproduce the observed dark
matter relic density. Motivated by this little drawback, in this paper we
consider an extension of the MDM scenario by three right-handed neutrinos. Two
specific choices for the MDM multiplet are studied: a fermionic $SU(2)_L$
quintuplet and a scalar $SU(2)_L$ septuplet. The lightest right-handed
neutrino, with tiny Yukawa couplings, never reaches thermal equilibrium in the
early universe and is produced by freeze-in. This creates a link between dark
matter and neutrino physics: dark matter can be non-thermally produced by the
decay of the lightest right-handed neutrino after freeze-out, allowing to lower
significantly the dark matter mass. We discuss the phenomenology of the
non-thermally produced MDM and, taking into account significant Sommerfeld
corrections, we find that the dark matter mass must have some specific values
in order not to be in conflict with the current bounds from gamma-ray
observations.
