Are there absolute value of field, which is not discrete in $\Bbb{R}_{＞0}^×$, and also not dense in $\Bbb{R}_{＞0}^×$?

Question

Let $K$ be a field with $\Bbb{Q}_p⊆K⊆\Bbb{C}_p$. Are there absolute value of field, which is not discrete in $\Bbb{R}_{＞0}^×$, and also not dense in $\Bbb{R}_{＞0}^×$?

All values I know is dense or discrete in $\Bbb{R}_{＞0}^×$, if you know or find some values which is neither dense nor discrete, I really appreciated it, thank you.

P.S 　I assume $K$ to be $\Bbb{Q}_p⊆K⊆\Bbb{C}_p$

Answer 1

No. In fact, any subgroup $G$ of $(\Bbb{R}^+, \cdot) \cong (\Bbb{R},+)$ is either discrete or dense. Additionally, $G$ is dicrete if and only if it is finitely generated.