Let $G$ be a locally profinite group and $K \le G$ an open subgroup. Does the restriction functor $\mathrm{Res}^G_K$ from the category of smooth $\mathbb{C}$-linear representations of $G$ to smooth $\mathbb{C}$-linear representations of $K$ preserve projective objects?

One is tempted to make an argument using the right adjoint $\mathrm{Ind}_K^G$, right exactness of which would suffice. But does the openness of $K$ imply the right exactness of the induction functor $\mathrm{Ind}_K^G$?