  We continue the explorations of derived \canal geometry started in [DAG-IX]
and in http://arxiv.org/abs/1506.09042. We describe the category of $\mathcal
O_X$-modules over a derived complex analytic space $X$ as the stabilization of
a suitable category of analytic algebras over $\mathcal O_X$. Finally, we apply
this description to introduce the notion of analytic square-zero extension and
prove a fundamental structure theorem for them.
