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The "fixed-points of actions on a $p$-acyclic space are $p$-acyclic" part of Smith theory easily extends to arbitrary $p$-groups. By induction on the order of the group: if $P$ acts on a $p$-acyclic $X$, choose a non-trivial proper normal subgroup $Z \leq P$ (these always exist; if $P$ is non-abelian take its centre), then $X^P = (X^Z)^{P/Z}$.
Let $U(1)$ act continuously on $D^n$, and $X(n)$ be the $\mathbb{Z}/p^n$ fixed points. This is a compact subset, and non-empty by Smith theory. Thus $X=\cap_{n=1}^\infty X(n)$ is also non-empty, by Cantor's intersection theorem. A point $x \in X$ is fixed by the subgroup of $U(1)$ of $p$-torsion p$-power-torsion points; this is dense, so$x$is fixed by the whole of$U(1)$. 1 Not quite your question, but I'll say it anyway. The "fixed-points of actions on a$p$-acyclic space are$p$-acyclic" part of Smith theory easily extends to arbitrary$p$-groups. By induction on the order of the group: if$P$acts on a$p$-acyclic$X$, choose a non-trivial proper normal subgroup$Z \leq P$(these always exist; if$P$is non-abelian take its centre), then$X^P = (X^Z)^{P/Z}$. Let$U(1)$act continuously on$D^n$, and$X(n)$be the$\mathbb{Z}/p^n$fixed points. This is a compact subset, and non-empty by Smith theory. Thus$X=\cap_{n=1}^\infty X(n)$is also non-empty, by Cantor's intersection theorem. A point$x \in X$is fixed by the subgroup of$U(1)$of$p$-torsion points; this is dense, so$x$is fixed by the whole of$U(1)\$.