This paper presents an online memory-augmented finite-sample model learning approach for uncertain nonlinear systems with prescribed-performance guarantees. Experience replay is leveraged to form a memory of events that have a significant effect on the performance of the learning mechanism, and the events in the memory are reused in the learning rule to guarantee that the modeling error converges to zero within a predefined settling time while remaining in a preselected prescribed bound during learning. An easy-to-check and verifiable metric defined on finite samples collected along the system's trajectories is provided to certify the prescribed-performance convergence. Finally, a simulation example verifies the efficiency of the proposed memory-augmented model learning approach.