The question is the same as in the title. More precisely, does there exist a **nonzero** smooth mapping $f : \mathbb{R} \to \mathbb{R}$ such that for all $k \in \mathbb{N}$, the $k$-th derivative $f^{(k)}$ satisfies $\sup_{t \in \mathbb{R}} \lvert f^{(k)}(t)\rvert \le 1$ (with $f^{(0)}$ being understood as $f$ itself), and $f$ itself is compactly supported?

I suspects the answer is affirmative, but I can not find a proof at the moment. Here is some elementary observations:

If we drop the requirement of compact support, then functions like $\sin(x)$ and $\cos(x)$, as well as suitable linear combinations of them, all satisfy this property. Perhaps one can construct examples out of these functions.

Perhaps one can mollify piecewise linear functions satisfying the bound condition for all derivatives except on a finite number of points.