  We prove that any projective coadmissible module over the locally analytic
distribution algebra of a compact $p$-adic Lie group is finitely generated. In
particular, the category of coadmissible modules does not have enough
projectives. In the Appendix a "generalized Robba ring" for uniform pro-$p$
groups is constructed which naturally contains the locally analytic
distribution algebra as a subring. The construction uses the theory of
generalized microlocalization of quasi-abelian normed algebras that is also
developed there. We equip this generalized Robba ring with a self-dual locally
convex topology extending the topology on the distribution algebra. This is
used to show some results on coadmissible modules.
