Let $K$ be a number field, and let $K((t))$ be the field of formal Laurent series. Let $p > 0$ be a prime.

I have two questions:

- Does there exist a discrete valuation subring $R$ of $K((t))$ of residue characteristic $p$ satisfying $\mathrm{Frac}(R) = K((t))$?
- Given a discrete valuation subring $A\subset K((t))$, is it always possible to find a discrete valuation subring $R$ dominating $A$ and satisfying $\mathrm{Frac}(R) = K((t))$?

(Of course (2) implies (1) since we may pick $A$ to be the localization of $\mathbb{Z}$ at $p$)