  Let $d$ be a (well-behaved) shortest-path metric defined on a path-connected
subset of $\mathbb{R}^2$ and let $\mathcal{D}=\{D_1,\ldots,D_n\}$ be a set of
geodesic disks with respect to the metric $d$. We prove that
$\mathcal{G}^{\times}(\mathcal{D})$, the intersection graph of the disks in
$\mathcal{D}$, has a clique-based separator consisting of
$O(n^{3/4+\varepsilon})$ cliques. This significantly extends the class of
objects whose intersection graphs have small clique-based separators.
  Our clique-based separator yields an algorithm for $q$-COLORING that runs in
time $2^{O(n^{3/4+\varepsilon})}$, assuming the boundaries of the disks $D_i$
can be computed in polynomial time. We also use our clique-based separator to
obtain a simple, efficient, and almost exact distance oracle for intersection
graphs of geodesic disks. Our distance oracle uses $O(n^{7/4+\varepsilon})$
storage and can report the hop distance between any two nodes in
$\mathcal{G}^{\times}(\mathcal{D})$ in $O(n^{3/4+\varepsilon})$ time, up to an
additive error of one. So far, distance oracles with an additive error of one
that use subquadratic storage and sublinear query time were not known for such
general graph classes.
