Aiming at the problems of strong model dependence, frequent controller actuation, and slow convergence of tracking errors in the repetitive operation of a class of unknown discrete-time nonlinear non-affine systems, this paper proposes an event-triggered data-driven finite-iteration learning control method. Based on iterative dynamic linearization, the proposed method constructs a dynamic linear data model in the iteration domain by using only the input and output data of the system. On this basis, an event-triggered control law, an iterative pseudo-partial derivative estimation law, and a fast learning control law with finite-iteration convergence are designed. Furthermore, by using the Lyapunov method, the finite-iteration convergence of the tracking error under a prescribed accuracy is proved, and a method for calculating the corresponding upper bound of the convergence iteration number is provided. The proposed method does not require any mechanism-based modeling or identification modeling process. It not only reduces the number of controller actuations, but also achieves finite-iteration convergence, enabling the tracking error to rapidly converge to a specified error bound within a finite number of iterations. The simulation study verifies the correctness of the theoretical results and the effectiveness of the proposed method.