Let $K$ be a local field of positive characteristic. I'm looking for an $K$ which satisfies the following condition.

・Value group of $K^{sep}$(Separable closure of $K$) is $\bigcup_{n≧1}(1/p^n) \Bbb{Z}$

$K$ should be like the form $ \Bbb{F}_q((t))$, so I need to determine $q$.