  We prove new cases of the Hilbert-Smith conjecture for actions by natural
homeomorphisms in symplectic topology. Specifically, we prove that the group of
$p$-adic integers $\mathbb Z_p$ does not admit non-trivial continuous actions
by Hamiltonian homeomorphisms, the $C^0$ limits of Hamiltonian diffeomorphisms,
on symplectically aspherical symplectic manifolds. For a class of symplectic
manifolds, including all standard symplectic tori, we deduce that a locally
compact group acting faithfully by homeomorphisms in the $C^0$ closure of
time-one maps of symplectic isotopies must be a Lie group. Our methods of proof
differ from prior approaches to the question and involve barcodes and power
operations in Floer cohomology. They also apply to other natural metrics in
symplectic topology, notably Hofer's metric. An appendix by Leonid Polterovich
uses this to deduce obstructions on Hamiltonian actions by semi-simple $p$-adic
analytic groups.
