The radio-frequency (RF) signal processing in real time is indispensable for advanced information systems, such as radar and communications. However, the latency performance of conventional processing paradigm is worsened by high-speed analog-to-digital conversion (ADC) generating massive data, and computation-intensive digital processing. Here, we propose to encode and process RF signals harnessing photonic spiking response in fully-analog domain. The dependence of photonic analog-to-spike encoding on threshold level and time constant is theoretically and experimentally investigated. For two classes of waveforms from real RF devices, the photonic spiking neuron exhibits distinct distributions of encoded spike numbers. In a waveform classification task, the photonic-spiking-based scheme achieves an accuracy of 92%, comparable to the K-nearest neighbor (KNN) digital algorithm for 94%, and the processing latency is reduced approximately from 0.7 s (code running time on a CPU platform) to 80 ns (light transmission delay) by more than one million times. It is anticipated that the asynchronous-encoding, and binary-output nature of photonic spiking response could pave the way to real-time RF signal processing.