We present a high-speed, energy-efficient Convolutional Neural Network (CNN) architecture utilising the capabilities of a unique class of devices known as analog Focal Plane Sensor Processors (FPSP), in which the sensor and the processor are embedded together on the same silicon chip. Unlike traditional vision systems, where the sensor array sends collected data to a separate processor for processing, FPSPs allow data to be processed on the imaging device itself. This unique architecture enables ultra-fast image processing and high energy efficiency, at the expense of limited processing resources and approximate computations. In this work, we show how to convert standard CNNs to FPSP code, and demonstrate a method of training networks to increase their robustness to analog computation errors. Our proposed architecture, coined AnalogNet, reaches a testing accuracy of 96.9% on the MNIST handwritten digits recognition task, at a speed of 2260 FPS, for a cost of 0.7 mJ per frame.