  We prove that if $(X,\mathsf d,\mathfrak m)$ is an essentially non-branching
metric measure space with $\mathfrak m(X)=1$, having Ricci curvature bounded
from below by $K$ and dimension bounded from above by $N \in (1,\infty)$,
understood as a synthetic condition called Measure-Contraction property, then a
sharp isoperimetric inequality \`a la L\'evy-Gromov holds true. Measure
theoretic rigidity is also obtained.
