Following the notion of $p$-adic analytic differential operators introduced by Ardakov--Wadsley, we establish a number of properties for coadmissible $\backslash wideparenD$-modules on rigid analytic spaces. Our main result is a $\backslash wideparenD$-module analogue of Kiehl's Proper Mapping Theorem, considering the 'naive' pushforward from $\backslash wideparenDX$-modules to $f\ast \backslash wideparenDX$-modules for proper morphisms $f:X\to Y$. Under assumptions which can be naturally interpreted as a certain properness condition on the cotangent bundle, we show that any coadmissible $\backslash wideparenDX$-module has coadmissible higher direct images. This implies among other things a purely geometric justification of the fact that the global sections functor in the rigid analytic Beilinson--Bernstein correspondence preserves coadmissibility, and we are able to extend this result to arbitrary twisted $\backslash wideparenD$-modules on analytified partial flag varieties. Our results rely heavily on the study of completed tensor products for $p$-adic Banach modules, for which we provide several new exactness criteria. We also show that the main results of Ardakov--Wadsley on the algebraic structure of $\backslash wideparenD$ still hold without assuming the existence of a smooth Lie lattice. For instance, we prove that the global sections $\backslash wideparenDX(X)$ form a Frechet--Stein algebra for any smooth affinoid $X$.