  For an operator-differential equation of the form $y^{(m)}(z) = Ay(z)$, where
$A$ is a closed linear operator on a Banach space over the the field of
$p$-adic numbers, the necessary and sufficient conditions on initial data for
the Cauchy problem to be well-posed in the class of locally analytic
vector-valued functions are given. The result is illustrated by $p$-adic
partial differential equations.
