Let $E\to M$ be a smooth vector bundle over a smooth manifold $M$.

Let $D$ be a differential operator defined on the space $\Gamma(E)$ of smooth sections of $E$. We fix a section $s\in \Gamma(E)$.

Assume that for every $x\in M$ there exist a natural number $n$ such that $D^n(s)(x)=0$.

Does this imply that $D^k(s)=0$, for some $k\in \mathbb{N}$?

If the answer is "No", what about if we assume that $D$ is an elliptic operator and $M$ is a compact manifold?

The motivation for this question is the following fact which I learned from the book of "R.P.Boas:A primer of real functions".

https://www.jstor.org/stable/10.4169/j.ctt5hh8x5

Fact: Assume that $f:\mathbb{R} \to \mathbb{R}$ is a smooth function with the property that for every $x\in \mathbb{R}$ there exist a natural number $n$ with $f^{(n)}(x)=0$.Then $f$ is a polynomial.