In this paper, we show that an eXtremely Large (XL) Multiple-Input Multiple-Output (MIMO) wireless system with appropriate analog combining components exhibits the properties of a universal function approximator, similar to a feedforward neural network. By treating the channel coefficients as the random nodes of a hidden layer and the receiver's analog combiner as a trainable output layer, we cast the XL MIMO system to the Extreme Learning Machine (ELM) framework, leading to a novel formulation for Over-The-Air (OTA) edge inference without requiring traditional digital processing nor pre-processing at the transmitter. Through theoretical analysis and numerical evaluation, we showcase that XL-MIMO-ELM enables near-instantaneous training and efficient classification, even in varying fading conditions, suggesting the paradigm shift of beyond massive MIMO systems as OTA artificial neural networks alongside their profound communications role. Compared to conventional ELMs and deep learning approaches, whose training takes seconds to minutes, the proposed framework achieves on par performance (above $90\%$ classification accuracy across multiple data sets) with optimization latency of few milliseconds under the same number of trainable parameters, considering rich fading, low noise channels with XL receive antennas, making it highly attractive for inference tasks with ultra-low-power devices.