We give an example of a totally disconnected set $E \subset {\mathbb R}^3$ which is not removable for quasiconformal homeomorphisms, i.e., there is a homeomorphism $f$ of ${\mathbb R}^3$ to itself which is quasiconformal off $E$, but not quasiconformal on all of ${\mathbb R}^3$. The set $E$ may be taken with Hausdorff dimension $2$. The construction also gives a non-removable set for locally biLipschitz homeomorphisms.