  Let $E$ be a field of characteristic $p$. The group $\mathbf{Z}_p^\times$
acts on $E((X))$ by $a \cdot f(X) = f((1+X)^a-1)$. This action extends to the
$X$-adic completion $\tilde{\mathbf{E}}$ of $\cup_{n \geq 0} E((X^{1/p^n}))$.
We show how to recover $E((X))$ from the valued $E$-vector space
$\tilde{\mathbf{E}}$ endowed with its action of $\mathbf{Z}_p^\times$. To do
this, we introduce the notion of super-H\"older vector in certain $E$-linear
representations of $\mathbf{Z}_p$. This is a characteristic $p$ analogue of the
notion of locally analytic vector in $p$-adic Banach representations of
$p$-adic Lie groups.
