Convolutional neural networks (CNNs) are widely used state-of-the-art computer vision tools that are becoming increasingly popular in high energy physics. In this paper, we attempt to understand the potential of CNNs for event classification in the NEXT experiment, which will search for neutrinoless double-beta decay in $^{136}$Xe. To do so, we demonstrate the usage of CNNs for the identification of electron-positron pair production events, which exhibit a topology similar to that of a neutrinoless double-beta decay event. These events were produced in the NEXT-White high-pressure xenon TPC using 2.6-MeV gamma rays from a $^{228}$Th calibration source. We train a network on Monte Carlo-simulated events and show that, by applying on-the-fly data augmentation, the network can be made robust against differences between simulation and data. The use of CNNs offer significant improvement in signal efficiency/background rejection when compared to previous non-CNN-based analyses.