In this paper, we study the existence of isometries among the composition operators between several Banach spaces of analytic functions on the unit disk in \(\mathbb {C}\) with particular focus on the case when the domain and target spaces are different. We find that there are no composition isometries between several classical Banach spaces and prove that the only composition isometries on the derivative Hardy space \(S^p\) are induced by rotations. Finally, we show that none of the spaces \(S^p_n\) consisting of the functions f whose n-th derivative is in the Hardy space \(H^p\) can support hypercyclic composition operators.
