<p>The nonlinear and memory-dependent behaviors of soft robots impose significant challenges to accurate dynamic modeling. This paper introduces a novel Mori-Zwanzig (MZ) Koopman operator framework that incorporates memory effects into the dynamic modeling of a cable-driven soft robotic arm. By integrating memory kernels from the MZ modal decomposition into the Koopman operator, the proposed approach renders a data-driven approximation of nonlinear memory-dependent dynamics by a linear dynamic model in the observable space. An experimental platform is employed to generate data for system identification with two scenarios examined: unidirectional bending and multidirectional deformation. The identified MZ-Koopman model is evaluated against the traditional Koopman model, linear state-space model, and LSTM neural-network model by comparing their predictions of the robotic arm’s motion against the truth data. The experimental results demonstrate that the MZ-Koopman model significantly improves modeling accuracy over the other three methods, particularly achieving more than 20% improvement over the traditional Koopman operator model. Additionally, this study explores the effects of actuation speed on memory-dependent behaviors and evaluates the impact of memory kernels on modeling accuracy.</p>