Let $K/\mathbb{Q}$ be a finite Galois extension, and $L/K$ an infinite Galois extension such that the Galois group $ \operatorname{Gal}(L/K) $ is isomorphic to a closed subgroup of $ \operatorname{GL}_{d}(\mathbb{Z}_{p}) $ for some positive integer $ d $ where $\mathbb{Z}_{p}$ is the ring of $p$-adic integers. Suppose that $L/\mathbb{Q}$ is not Galois, and let $M$ denote the Galois closure of $L/\mathbb{Q}$.

My question is the following:

1. Is the ramification index of the Galois extension $M/L$ finite for any prime ideal of $L$?
2. Is the Galois group $ \operatorname{Gal}(M/L) $ isomorphic to a closed subgroup of $ \operatorname{GL}_{m}(\mathbb{Z}_{p}) $ for some positive integer $ m $?

For example, if $M/L$ is finite, then the answer to the above question is Yes. I don't know if $M/L$ is always finite.