Suppose $K$ is an extension of $\mathbb{Q}_{2}$ (I could ask this for any $p$, but I'm especially interested in $p = 2$) which contains the maximal unramified extension $\mathbb{Q}_{2}^{\mathrm{unr}}$ and also contains all roots of unity (including $2$-power ones!) Then does $K$ contain all roots of all elements of $\mathcal{O}_{K}^{\times}$?

This seems to boil down to knowing the ramification behavior of extensions obtained by adjoining roots of units in $\mathcal{O}_{K}^{\times}$. I feel like this should be hidden somewhere in the main results of class field theory, but I'm not sure.