Is the ring of formal power series a localization of some non local ring at prime ideals?

Question

Consider $F = K[[x]],$ the formal series over field $K.$ We know that $K$ is a local ring with maximal ideal $(x).$ Does there exist a non local ring $R$ and prime ideal $p$ such that $R_p = K[[x]]?$ Since $(x)$ is the only prime ideal in $F,$ $p$ should be minimal prime ideal in $R.$ I tried to polynomia ring $R=K[x]$ with $p=(x)$ the corresponding localisation will not be $F.$ I am not able to find an example or show that no such $R$ exist.