Exercise 4, § 10

Let $K$ be a quadratic extension of $\mathbf{Q}_p$ and let $\omega \in K - \mathbf{Q}_p$. Let $G = \mathbf{Q}_p + \mathbf{Z}_p \omega$, considered as a Lie group over K. The only automorphisms of G are the $(x, y) \mapsto (\lambda x, y)$ with $\lambda$ invertible in $\mathbf{Z}_p$. The set of these automorphisms cannot be endowed with a Lie group structure over K possessing the properties of Theorem 1.