Let K be a complete non-archimedean field (say $\mathbb{C_p}$) and let G be a discrete group. Since {e} is an open p-compatible compact subgroup of G, G admits a (left) K-valued Haar measure $\mu$. Obviously $\mu(\{g\})=\mu(\{e\})$ for all $g\in G$. But now there is a canonical isometry of the space of (tight) measures M(G) and c_{0}(G). Thus, the set of the points with $|\mu(\{x\})|\ge |\mu(\{e\})|$ must be compact. Thus, the only discrete groups with nontrivial Haar measure are finite. What am I (possibly) doing wrong.

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