Exercise 2, § 9

Suppose that K is ultrametric. Let G be a finite-dimensional Lie group, $\mathfrak{g}$ its Lie algebra and A a finite subset of G. Then $Z_G(A)$ is a Lie subgroup of G with Lie algebra $\mathfrak{z}_\mathfrak{g}(A)$. (Argue as in Proposition 8.)