Let $H(D)$ be the set of all analytic functions on the open unit disc $D$ and $H^{\infty} (D)$ the Banach algebra of all bounded analytic functions on $D$. We characterize surjective, not necessarily linear, isometries on $S^{\infty} = \{ f \in H(D) : f' \in H^{\infty}(D) \}$.