Brunnian links of 3-balls in the 4-sphere

For each integer $n\ge 2$, we construct infinitely many $n$-component Brunnian links of 3-balls in $S^4$. Our main tool is the third author’s result on the existence of splitting spheres for the trivial two-component link of $2$-spheres in $S^4$; we also give a new proof of this.