  We establish a Lefschetz hyperplane theorem for the Berkovich
analytifications of Jacobians of curves over an algebraically closed
non-Archimedean field. Let $J$ be the Jacobian of a curve $X$, and let $W_d
\subset J$ be the locus of effective divisor classes of degree $d$. We show
that the pair $(J^{an},W_d^{an})$ is $d$-connected, and thus in particular the
inclusion of the analytification of the theta divisor $\Theta^{an}$ into
$J^{an}$ satisfies a Lefschetz hyperplane theorem for $\mathbb{Z}$-cohomology
groups and homotopy groups. A key ingredient in our proof is a generalization,
over arbitrary characteristics and allowing arbitrary singularities on the
base, of a result of Brown and Foster for the homotopy type of analytic
projective bundles.
