Does there exist an inclusion map making a given boundary point into an interior point?

Question

Every smooth manifold is assumed to be Hausdorff and second-countable.

Let $M$ be a smooth manifold with nonempty boundary.

$\forall p\in \partial M$, does there exist a smooth manifold $N$ with or without boundary such that $M\subset N$ is an embedded submanifold with boundary, $M$ has the same dimension as $N$, and $p$ is an interior point of $N$?