Let $G={\rm Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ be the absolute Galois group over $\mathbb{Q}$ and let $\mathbb{Q}_{p}$ be the $p$-adic field. It is well-known that the image of a continuous group homomorphism $\rho: G\to {\rm GL}_n(\mathbb{Q}_p)$ is a Lie subgroup of ${\rm GL}_n(\mathbb{Q}_p)$ in the obvious sense. As far as I know, the proof depends on Cartan's closed-subgroup theorem, which says that a topologically closed subgroup of an analytic group is again analytic.
However, if $K$ is a local field of positive characteristic, the above theorem of Cartan is not true in general, see for example the following paper
https://arxiv.org/pdf/2203.15861.pdf
So my question is: Let $L$ be a function field over a finite field and let $G={\rm Gal}(L^{sep}/L)$ be the Galois group of the separable closure of $L$. Take $K$ as another function field of the same (positive) characteristic as $L$. Take $\lambda$ to be a prime place of $K$ so that the induced field $K_{\lambda}$ is a local field that is complete with respect to its natural ultrametric. Suppose $\rho: G\to {\rm GL}_n(K_{\lambda})$ be a continuous group homomorphism. Then is the image $\rho(G)$ still a Lie subgroup of ${\rm GL}_n(K_{\lambda})$?
There is a natural map $$Gal(\Bbb{F}_p(x)^{sep}/\Bbb{F}_p(x))\to Gal(\Bbb{F}_{\displaystyle p^{p^\infty}}/\Bbb{F}_p) \to \Bbb{Z}_p \to (1+x^p)^{\Bbb{Z}_p}$$ $$ =1+x^p \Bbb{F}_p[[x^p]]\subset GL_1(\Bbb{F}_p((x)))$$ $a\mapsto 1+a$ is a chart $x \Bbb{F}_p[[x]]\to 1+x \Bbb{F}_p[[x]]$ (multiplication and inverse are given by analytic functions) making $GL_1(\Bbb{F}_p((x)))$ a Lie group.
And $1+x^p \Bbb{F}_p[[x^p]]$ isn't a Lie subgroup,
in the same way that $\Bbb{R}$ isn't a Lie subgroup of $\Bbb{C}$. Or maybe should we change our definition of Lie subgroup?