Has the Pontryagin dual of a module the same length of the original module?

Question

Let $M$ be a topological module over a complete noetherian local commutative ring $R$ of finite residue characteristic $p$. Its Pontryagin dual is defined to be $M^\vee:=Hom_{cont}(M,\mathbb{Q}_p/\mathbb{Z}_p)$.

Is it true that the length of $M$ as an $R$-module coincides with the length of $M^\vee$ as an $R$-module?