The (unsolved) Hilbert-Smith conjecture states that any locally compact group acting faithfully on a manifold has to be a Lie group: http://en.wikipedia.org/wiki/Hilbert%E2%80%93Smith_conjecture
However, it turns out that it is enough to prove this for $\mathbb{Z}_p$, and the conjecture follows proving that $\mathbb{Z}_p$ has no continuous faithful action on a manifold.
