Methods have been proposed in recent years aimed at pushing photoacoustic imaging resolution beyond the acoustic diffraction limit, among which those based on random speckle illumination show particular promise. In this Letter, we propose a data-driven deep learning approach to processing the added spatiotemporal information resulting from speckle illumination, where the neural network learns the distribution of absorbers from a series of different samplings of the imaged area. In ex-vivo experiments based on the tomography configuration with prominent artifacts, our method successfully breaks the acoustic diffraction limit and delivers better results in identifying individual targets when compared against a selection of other leading methods.