Weyl semimetals (WSMs) are gapless topological states of matter with broken inversion and/or time reversal symmetry. WSMs can support a circulating photocurrent when illuminated by circularly polarized light at normal incidence. Here, we report a spatially dispersive circular photogalvanic effect (s-CPGE) in a WSM that occurs with a spatially varying beam profile. Our analysis shows that the s-CPGE is controlled by a symmetry selection rule combined with asymmetric carrier excitation and relaxation dynamics. By evaluating the s-CPGE for a minimal model of a WSM, a frequency-dependent scaling behaviour of the photocurrent is obtained. Wavelength-dependent measurements from the visible to mid-infrared range show evidence of Berry curvature singularities and band inversion in the s-CPGE response. We present the s-CPGE as a promising spectroscopic probe for topological band properties, with the potential for controlling photoresponse by patterning optical fields on topological materials to store, manipulate and transmit information. Ministry of Education (MOE) National Research Foundation (NRF) Accepted version RA acknowledges the support from the Office of Naval Research MURI (grant #N00014-17-1-2661) and the US Army Research Office (grant # W911NF-17-1-0436). Work by EJM and ZA was supported by the Department of Energy (grant# DE FG02 84ER45118). The crystal growth effort (PY and ZL) was supported by the Singapore National Research Foundation under NRF RF Award No. NRF-RF2013-08 and Tier 2 MOE2016-T2-2-153. RA, EJM, CLK and AMR acknowledge the support from the Center of Excellence for Materials Research and Innovation Seed Grant and the MRSEC iSuperSEED Supplement.