Erdős Problem #229 — Entire Functions with Prescribed Derivative Zeros

Let $(S_n)_{n\in\mathbb{N}}$ be a sequence of subsets of $\mathbb{C}$ such that the derived set of each $S_n$ is empty, meaning that no $S_n$ has a finite limit point. Does there exist a transcendental entire function $f:\mathbb{C}\to\mathbb{C}$ such that, for every integer $n\geq 1$, there is an integer $k\geq 0$ for which

$$f^{(k)}(z)=0$$

for every $z\in S_n$?