Vertical-Cavity Surface-Emitting Lasers (VCSELs) are highly promising devices for the construction of neuromorphic photonic information processing systems, due to their numerous desirable properties such as low power consumption, high modulation speed, compactness, and ease of manufacturing. Of particular interest is the ability of VCSELs to exhibit neurallike spiking responses, much like biological neurons, but at ultrafast sub-nanosecond rates; thus offering great prospects for high-speed light-enabled neuromorphic (spike-based) processors. Recent works have shown the use the spiking dynamics in VC-SELs for pattern recognition and image processing problems such as image data encoding and edge-feature detection. Additionally, VCSELs have also been used recently as nonlinear elements in photonic reservoir computing (RC) implementations, yielding excellent state of the art operation. This work introduces and experimentally demonstrates for the first time the new concept of a Ghz-rate photonic spiking neural network (SNN) built with a single VCSEL neuron. The reported system effectively implements a photonic VCSEL-based spiking reservoir computer, and demonstrates its successful application to a complex nonlinear classification task. Importantly, the proposed system benefits from a highly hardware-friendly, inexpensive realization (built with a single VCSEL and off-the-shelf fibre-optic components), for high-speed (GHz-rate inputs) and low-power (sub-mW optical input power) photonic operation. These results open new pathways towards future neuromorphic photonic spikebased information processing systems based upon VCSELs (or other laser types) for novel ultrafast machine learning and AI hardware.