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As deep neural networks (DNNs) grow to solve increasingly complex problems, they are becoming limited by the latency and power consumption of existing digital processors. For improved speed and energy efficiency, specialized analog optical and electronic hardware has been proposed, however, with limited scalability (input vector length $K$ of hundreds of elements). Here, we present a scalable, single-shot-per-layer analog optical processor that uses free-space optics to reconfigurably distribute an input vector and integrated optoelectronics for static, updatable weighting and the nonlinearity -- with $K \approx 1,000$ and beyond. We experimentally test classification accuracy of the MNIST handwritten digit dataset, achieving 94.7% (ground truth 96.3%) without data preprocessing or retraining on the hardware. We also determine the fundamental upper bound on throughput ($\sim$0.9 exaMAC/s), set by the maximum optical bandwidth before significant increase in error. Our combination of wide spectral and spatial bandwidths in a CMOS-compatible system enables highly efficient computing for next-generation DNNs.