Word norm in compact subsets of finitely generated groups

Question

Let $G$ be a finitely generated topological group, not necessarily discrete. Fix a finite generating set $S$ and denote by $|x|$ the word norm of $x \in G$ with respect to this generating set, i.e., the minimal $n \in \mathbb{N}$ such that there exist $s_1, \ldots, s_n \in S$ with $x = s_1 \cdots s_n$. Let $K$ be a compact subset of $G$. Is it true that $\{ |k| : k \in K \} \subseteq \mathbb{N}$ is bounded?

Answer 1

If I understood the question correctly, no.

Pick $G=\mathbb{Z}$, $S=\{ 1 \}$, but endow $G$ with the $p$-adic topology (for example $p=2$). Then $K=\{p^k\}_{k \geq 0} \cup \{0\}$ is a converging sequence and its endpoint, so is a compact subset, and clearly the lengths are not bounded on $K$.