  Let $G\subset x{\mathbb F}_q[\![x]\!]$ ($q$ is a power of the prime $p$) be a
subset of formal power series over a finite field such that it forms a compact
abelian $p$-adic Lie group of dimension $d\ge 1$. We establish a necessary and
sufficient condition for the APF extension of local field corresponding to
$\left({\mathbb F}_q(\!(x)\!), G\right)$ under the field of norms functor to be
an extension of $p$-adic fields. We then apply this result to study family of
invertible power series with coefficients in a $p$-adic integers ring and
commute with a fixed noninvertible power series under the composition of power
series.
