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Let $D$ be a bounded domain in a complex Banach space. According to theEarle-Hamilton fixed point theorem, if a holomorphic mapping $F : D \mapsto D$maps $D$ strictly into itself, then it has a unique fixed point and itsiterates converge to this fixed point locally uniformly. Now let $\mathcal{B}$be the open unit ball in a complex Hilbert space and let $F : \mathcal{B}\mapsto \mathcal{B}$ be holomorphic. We show that a similar conclusion holdseven if the image $F(\mathcal{B})$ is not strictly inside $\mathcal{B}$, but iscontained in a horosphere internally tangent to the boundary of $\mathcal{B}$.This geometric condition is equivalent to the fact that $F$ is asymptoticallystrongly nonexpansive with respect to the hyperbolic metric in $\mathcal{B}$.

A generalized version of the Earle-Hamilton fixed point theorem for the Hilbert ball