  We show that there exist split, orientable, 2-component surface-links in
$S^4$ with non-isotopic splitting spheres in their complements. In particular,
for non-negative integers $m,n$ with $m\ge 4$, the unlink $L_{m,n}$ consisting
of one component of genus $m$ and one component of genus $n$ contains in its
complement two smooth splitting spheres that are not topologically isotopic in
$S^4\setminus L_{m,n}$. This contrasts with link theory in the classical
dimension, as any two splitting spheres in the complement of a 2-component
split link $L\subset S^3$ are isotopic in $S^3\setminus L$.

Geometric Topology