Common fixed points of commuting holomorphic maps in the unit ball of ${\mathbb C}^n$

Let $\mathbb B^n$ be the unit ball of $\mathbb{C}^n$ ($n>1$). We prove that if $f,g \in \operatorname{Hol}(\mathbb B^n,\mathbb B^n)$ are holomorphic self-maps of $\mathbb B^n$ such that $f \circ g = g \circ f$, then $f$ and $g$ have a common fixed point (possibly at the boundary, in the sense of $K$-limits). Furthermore, if $f$ and $g$ have no fixed points in $\mathbb B^n$, then they have the same Wolff point, unless the restrictions of $f$ and $g$ to the one-dimensional complex affine subset of $\mathbb B^n$ determined by the Wolff points of $f$ and $g$ are commuting hyperbolic automorphisms of that subset.