It follows from Whitney extension theorem that for every closed set $ C \subseteq \mathbb{R}^n $ and for every $ k \geq 1 $ there exists a function $ f \in C^k(\mathbb{R}^n) $ such that $ C = \{x : f(x)=0 \} $ and $ D^if(x) =0 $ for every $ x \in C $ and $ i = 1, \ldots , k $.
Is it possible to replace $ k $ with $ \infty $ in the statement above?
Yes, for example, take an open cover of the complement of $C$ by countably many open balls of radius $r_i$ centered at $v_i$, and use a partition of unity $\{f_i\}_{i∈I}$ subordinate to this cover to glue bump functions $\exp(-(\max(r_i^2-‖x-v_i‖^2,0))^{-2})$ into a globally smooth function $$∑_{i∈I}f_i \exp(-(\max(r_i^2-‖x-v_i‖^2,0))^{-2}).$$
This can be done iteratively. Choose a cover of the complement of $C$ by a countable family of open balls $B_n$. For each of the open balls, consider a smooth function $f_n$ such that $\lbrace f_n>0\rbrace=B_n$. Then for $\varepsilon_n$ going to zero fast enough, $\sum_n\varepsilon_nf_n$ and all its derivatives converges uniformly, so the limit exists, is smooth, and the derivatives are the sum are the sum of the derivatives. In particular, they all vanish on $C$.
I give some details in this answer in the more general case of Hilbert spaces.
