This paper treats a holomorphic self-mapping $f: \Omega \rightarrow \Omega$ of a bounded domain $\Omega$ in a separable Hilbert space ${\cal H}$ with a fixed point $p$. In case the domain is convex, we prove an infinite-dimensional version of the Cartan-Carath\'eodory-Kaup-Wu Theorem. This is basically a rigidity result in the vein of the uniqueness part of the classical Schwarz lemma. The main technique, inspired by an old idea of H. Cartan, is iteration of the mapping $f$ and its derivative. A normality result for holomorphic mappings in the compact-weak-open topology, due to Kim and Krantz, is used.
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