Compact subset of a t.d.l.c. group contained in compact-open subgroup?

Question

Let $G$ be a totally disconnected, locally compact group which is the union of its compact-open subgroups (or perhaps only $\sigma$-compact), and let $C \subset G$ be a compact set. Can one conclude that $C$ is contained in some compact-open subgroup of $G$?

Answer 1

The famous Burnside problem asks whether a finitely generated group, whose elements all have finite order, must be a finite group.

This question was answered in the negative in 1964 by Golod and Shafarevich.

If $G$ is a counter-example to the Burnside problem, once it is equipped with the discrete topology, it is also a counter-example to the question posed by the OP. The finite generating set is a compact set not contained in any compact group!