Digital signal processing has revolutionized many fields of science and engineering, but it still shows critical limits, mainly related to the complexity, power consumption, and limited speed of analogue-to-digital converters. A long-sought solution to overcome these hurdles is optical analog computing. In this regard, flat optics has been recently unveiled as a powerful platform to perform data processing in real-time, with low power consumption and a small footprint. So far, these explorations have been mainly limited to linear optics. Arguably, significantly more impact may be garnered from pushing this operation towards nonlinear processing of the incoming signals. In this context, we demonstrate here that nonlinear phenomena combined with engineered nonlocality in flat optics devices can be leveraged to synthesize Volterra kernels able to outperform linear optical analog image processing.