  We reconsider the recently proposed mimetic gravity, focusing in particular
on whether the theory is able to reproduce the inferred flat rotation curves of
galaxies. We extend the theory by adding a non-minimal coupling between matter
and mimetic field. Such coupling leads to the appearance of an extra force
which renders the motion of test particles non-geodesic. By studying the weak
field limit of the resulting equations of motion, we demonstrate that in the
Newtonian limit the acceleration law induced by the non-minimal coupling
reduces to a Modified Newtonian Dynamics (MOND)-like one. In this way, it is
possible to reproduce the successes of MOND, namely the explanation for the
flat galactic rotation curves and the Tully-Fisher relation, within the
framework of mimetic gravity, without the need for particle dark matter. The
scale-dependence of the recovered acceleration scale opens up the possibility
of addressing the missing mass problem not only on galactic but also on cluster
scales: we defer a full study of this issue, together with a complete analysis
of fits to spiral galaxy rotation curves, to an upcoming companion paper.
