In the last few decades, it has been proven that ground-based experiments devoted to the study of ultra-high-energy (UHE) cosmic rays are also powerful tools in the quest for UHE photons. The search for these elusive particles, never unambiguously detected so far, relies on the capability of distinguishing air showers initiated by primary photons among the background of the ones generated by nuclei. In this study, we explore the possibility of exploiting an array of water-Cherenkov detectors to distinguish photon-induced air showers based on the shape of the signal traces recorded by the individual detector stations of the array, using the experimental setup of the Pierre Auger Observatory as an example. A photon-induced air shower is dominated by its electromagnetic component, which tends to reach the ground later and to be more spread in time than the muonic one. Maximizing the discrimination power by considering the whole time evolution of the signals implies dealing with hundreds of variables. Additionally, we have to take into account other dependences of the signal shape, such as the energy and zenith angle of the primary particle and the distance of each station from the shower core. For such reasons, we make use of deep neural networks. Here we explore a Convolutional Neural Network algorithm and test it on air shower simulations. We show that, thanks to this innovative approach, it is possible to reach high levels of accuracy in classifying simulated air shower events, providing a promising tool to distinguish UHE photons.