Let $V \rightarrow M$ be a smooth vector bundle over a smooth compact manifold $M$ (without boundary) and $X \subset M$ a smooth submanifold of $M$, that is not necessarily closed. Suppose $s: X \rightarrow V$ is a smooth section, that extends continuously to $\bar{X}$, the closure of $X$ inside $M$. Does there always exist a smooth section $\tilde{s}: M \rightarrow V$, that restricted to $\bar{X}$ is equal to $s$?

Note that, if I replaced the word smooth with continuous the answer is yes. We can construct such a section using a continuous partition of unity and using the compactness of $\bar{X}$. And similarly, if $X$ was compact then there exists a smooth extension of $s$, this time using a smooth partition of unity and also using the compactness of $X$.