As artificial neural networks (ANNs) continue to make strides in wide-ranging and diverse fields of technology, the search for more efficient hardware implementations beyond conventional electronics is gaining traction. In particular, optical implementations potentially offer extraordinary gains in terms of speed and reduced energy consumption due to the intrinsic parallelism of free-space optics. At the same time, a physical nonlinearity—a crucial ingredient of an ANN—is not easy to realize in free-space optics, which restricts the potential of this platform. This problem is further exacerbated by the need to also perform the nonlinear activation in parallel for each data point to preserve the benefit of linear free-space optics. Here, we present a free-space optical ANN with diffraction-based linear weight summation and nonlinear activation enabled by the saturable absorption of thermal atoms. We demonstrate, via both simulation and experiment, image classification of handwritten digits using only a single layer and observed 6% improvement in classification accuracy due to the optical nonlinearity compared to a linear model. Our platform preserves the massive parallelism of free-space optics even with physical nonlinearity, and thus opens the way for novel designs and wider deployment of optical ANNs.