Deep neural networks with applications from computer vision to medical diagnosis^{, , , –} are commonly implemented using clock-based processors^{, , , , , , , –}, in which computation speed is mainly limited by the clock frequency and the memory access time. In the optical domain, despite advances in photonic computation^{, –}, the lack of scalable on-chip optical non-linearity and the loss of photonic devices limit the scalability of optical deep networks. Here we report an integrated end-to-end photonic deep neural network (PDNN) that performs sub-nanosecond image classification through direct processing of the optical waves impinging on the on-chip pixel array as they propagate through layers of neurons. In each neuron, linear computation is performed optically and the non-linear activation function is realized opto-electronically, allowing a classification time of under 570 ps, which is comparable with a single clock cycle of state-of-the-art digital platforms. A uniformly distributed supply light provides the same per-neuron optical output range, allowing scalability to large-scale PDNNs. Two-class and four-class classification of handwritten letters with accuracies higher than 93.8% and 89.8%, respectively, is demonstrated. Direct, clock-less processing of optical data eliminates analogue-to-digital conversion and the requirement for a large memory module, allowing faster and more energy efficient neural networks for the next generations of deep learning systems.