Event-based cameras are becoming a popular solution for efficient, low-power eye tracking. Due to the sparse and asynchronous nature of event data, they require less processing power and offer latencies in the microsecond range. However, many existing solutions are limited to validation on powerful GPUs, with no deployment on real embedded devices. In this paper, we present EETnet, a convolutional neural network designed for eye tracking using purely event-based data, capable of running on microcontrollers with limited resources. Additionally, we outline a methodology to train, evaluate, and quantize the network using a public dataset. Finally, we propose two versions of the architecture: a classification model that detects the pupil on a grid superimposed on the original image, and a regression model that operates at the pixel level.