Consider a locally profinite group $G$, i.e. a locally compact, totally disconnected topological group. Suppose it admits an open maximal compact subgroup named $K$. It is known that $G$ admits as a neighborhood basis of the identity element a collection $\{ K_i \}$ of open compact subgroups, but what can we say about the countability of this family?

In the basic examples I know, $GL_1(\mathbb{Q}_p)$ and $GL_2(\mathbb{Q}_p)$, the family ${K_i}$ is in fact countable, is it always true? Any reference is greatly appreciated, thanks.