Let $X$ be a Hausdorff totally disconnected space and $G$ be a compact (or even profinite) group acting continuously on $X$. Is it true that the orbit of any closed subset $Y$ under $G$ is closed in $X$?
The main example I have on my head is $X = \mathbb{Z} _p / \{0\}$ and $G = \mathbb{Z}_p^\times$ acting by multiplication, and in this case it is true mainly because it is the restriction of an action of $G$ on the profinite space $X \cup \{0\}$, and both $X$ and $\{0\}$ are $G$-equivariant. I'm interested in understanding if this holds in more generality.