  Let $K$ be a complete non-archimedean valuation field of characteristic $0$,
with non-trivial valuation, equipped with (possibly multiple) commuting bounded
derivations. We prove a decomposition theorem for finite differential modules
over $K$, where decompositions regarding the extrinsic subsidiary
$\partial$-generic radii of convergence in the sense of Kedlaya-Xiao. Our
result is a refinement of a previous decomposition theorem due to Kedlaya and
Xiao. As a key step in the proof, we prove a decomposition theorem in a
stronger form in the case where $K$ is equipped with a single derivation. To
achieve this goal, we construct an object $f_{0*}L_0$ representing the usual
dual functor and study some filtrations of $f_{0*}L_0$, which is used to
construct the direct summands appearing in our decomposition theorem.
