Let $p$ be a prime. There exist following containment : $$\mathbb{Q}_p \subset \mathbb{Q}_p^{\rm nr} \subset \mathbb{Q}_p^{\rm tr} \subset \overline{\mathbb{Q}_p}$$ Here $\mathbb{Q}_p^{\rm nr}$ and $\mathbb{Q}_p^{\rm tr}$ are the maximal unramified extension and the maximal tamely ramified extension over $\mathbb{Q}_p$ respectively.

Let $k$ be a Galois extension over $\mathbb{Q}_p$. We denote $k \cap \mathbb{Q}_p^{\rm nr}$ by $k_{\rm nr}$ and $k \cap \mathbb{Q}_p^{\rm tr}$ by $k_{\rm tr}$.

Question : Does there exist any infinite Galois extension $k$ over $\mathbb{Q}_p$ such that ${\rm Gal}(k_{\rm nr} / \mathbb{Q}_p)$ and ${\rm Gal} (k / k_{\rm tr})$ is finite?

Clearly, such $k$ is not an abelian extension over $\mathbb{Q}_p$. Though I doubt its existence after looking this answer and this article, I can't conclude rigorously.

P. S. If we change $\mathbb{Q}_p$ by any local field $F$, what more can be told?

Let $p$ be a prime. There exist following containment : $$\mathbb{Q}_p \subset \mathbb{Q}_p^{\rm nr} \subset \mathbb{Q}_p^{\rm tr} \subset \overline{\mathbb{Q}}_p$$ Here $\mathbb{Q}_p^{\rm nr}$ and $\mathbb{Q}_p^{\rm tr}$ are the maximal unramified extension and the maximal tamely ramified extension over $\mathbb{Q}_p$ respectively.

Let $k$ be a Galois extension over $\mathbb{Q}_p$. We denote $k \cap \mathbb{Q}_p^{\rm nr}$ by $k_{\rm nr}$ and $k \cap \mathbb{Q}_p^{\rm tr}$ by $k_{\rm tr}$.

Question : Does there exist any infinite Galois extension $k$ over $\mathbb{Q}_p$ such that ${\rm Gal}(k_{\rm nr} / \mathbb{Q}_p)$ and ${\rm Gal} (k / k_{\rm tr})$ is finite?

Clearly, such $k$ is not an abelian extension over $\mathbb{Q}_p$. Though I doubt its existence after looking this answer and this article, I can't conclude rigorously.

P. S. If we change $\mathbb{Q}_p$ by any local field $F$, what more can be told?