  The classical Julia-Wolff-Carath{\'e}odory Theorem characterizes the
behaviour of the derivative of an analytic self-map of a unit disc or of a
half-plane of the complex plane at certain boundary points. We prove a version
of this result that applies to noncommutative self-maps of noncommutative
half-planes in von Neumann algebras at points of the distinguished boundary of
the domain. Our result, somehow surprisingly, relies almost entirely on simple
geometric properties of noncommutative half-planes, which are quite similar to
certain geometric properties of classical hyperbolic spaces.
