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Imaging flow cytometry systems play a critical role in the identification and characterization of large populations of cells or micro-particles. Such systems typically leverage deep artificial neural networks to classify samples. Here we show that an event-based camera and neuromorphic processor can be used in a flow cytometry setup to solve a binary particle classification task with less memory usage, and promising improvements in latency and energy scaling. To reduce the complexity of the spiking neural network, we combine the event-based camera with a free-space optical setup which acts as a non-linear high-dimensional feature map that is computed at the speed of light before the event-based camera receives the signal. We demonstrate, for the first time, a spiking neural network running on neuromorphic hardware for a fully event-based flow cytometry pipeline with 98.45% testing accuracy. Our best artificial neural network on frames of the same data reaches only 97.51%, establishing a neuromorphic advantage also in classification accuracy. We further show that our system will scale favorably to more complex classification tasks. We pave the way for real-time classification with throughput of up to 1,000 samples per second and open up new possibilities for online and on-chip learning in flow cytometry applications.