This paper justifies an assertion in

(Elder, Proc AMS 137 (2009), no 4, 1193--1203) that Galois scaffolds make the
questions of Galois module structure tractable. Let $k$ be a perfect field of
characteristic $p$ and let $K=k((T))$. For the class of characteristic $p$
elementary abelian $p$-extensions $L/K$ with Galois scaffolds described in
mentioned paper, we give a necessary and sufficient condition for the valuation
ring $\mathfrak{O}_L$ to be free over its associated order $\mathcal{A}_{L/K}$
in $K[\Gal(L/K)]$. Interestingly, this condition agrees with the condition
found by Y. Miyata, concerning a class of cyclic Kummer extensions in
characteristic zero.