  We extend Urban's construction of eigenvarieties for reductive groups $G$
such that $G(\mathbb{R})$ has discrete series to include characteristic $p$
points at the boundary of weight space. In order to perform this construction,
we define a notion of "locally analytic" functions and distributions on a
locally $\mathbb{Q}_p$-analytic manifold taking values in a complete Tate
$\mathbb{Z}_p$-algebra in which $p$ is not necessarily invertible. Our
definition agrees with the definition of locally analytic distributions on
$p$-adic Lie groups given by Johansson and Newton.
