  Let $G$ be a $p$-divisible group over a complete discrete valuation ring $R$
of characteristic $p$. The generic fiber of $G$ determines a Galois
representation $\rho$. The image of $\rho$ admits a ramification filtration and
a Lie filtration. We relate these filtrations in the case $G$ is one
dimensional, giving an equicharacteristic version of Sen's theorem in this
setting. This result generalizes results of Gross and Chai. Additionally, we
prove that the representation associated to the \'etale part of $G$ is
irreducible.
