To tackle the path-following problem of marine floating bodies with strong nonlinear dynamics, this paper proposes a model predictive control (MPC) scheme based on an online predictor using Koopman operator learning. By introducing delayed states and corresponding derivatives into the observable function, the system dynamics are lifted to Hilbert space, where the linear model of marine floating body is more accurate. Thereby, a linear predictor is designed based on the Koopman operator, in which this data-driven approach relies on nonlinear transformation of system input-output measurements by solving the least squares problem in the lifting space. Then, the linear model obtained from the extended dynamic mode decomposition is used as the internal model of the MPC controller. Therefore, the MPC optimization problem designed in this way has the same computational complexity as the one in the linear MPC scenario. Moreover, both linear inequality constraints and nonlinear constraints arising from the dynamic characteristics of marine floating bodies can be imposed in a linear manner on the state and control inputs. To reduce computation time, an incremental update strategy is used by the online predictor. The proposed control scheme is applied to the path-following of a marine floating body, and extensive experiments are conducted to verify its effectiveness.