  The affinoid envelope, $\widehat{U(\mathcal{L})}$ of a free, finitely
generated $\mathbb{Z}_p$-Lie algebra $\mathcal{L}$ has proven to be useful
within the representation theory of compact $p$-adic Lie groups. Our aim is to
further understand the algebraic structure of $\widehat{U(\mathcal{L})}$, and
to this end, we will define a Dixmier module over $\widehat{U(\mathcal{L})}$,
and prove that this object is generally irreducible in case where $\mathcal{L}$
is nilpotent. Ultimately, we will prove that all primitive ideals in the
affinoid envelope can be described in terms of the annihilators of Dixmier
modules, and using this, we aim towards proving that these algebras satisfy a
version of the classical Dixmier-Moeglin equivalence.
