Basalt fibre-reinforced polymer (BFRP) composites probably tend to replace some carbon fibre-reinforced polymer (CFRP) applications due to their excellent specific strengths and sustainability. Despite the published early promising results concerning the material properties of BFRP, their application is not widespread, and their machinability is not supported widely by published experiences. The main aim of the present study is to experimentally investigate the drilling-induced geometrical damages of BFRP and CFRP composites. Drilling experiments were conducted at various feed and cutting speed levels using a solid carbide twist drill. The drilling-induced burr was analysed by a Mitutoyo 361–804 digital microscope, a Mitutoyo SJ400 surface tester recorded the surface roughness, and the microstructure was analysed by a Zeiss Evo MA 10 scanning electron microscope. The measured data were evaluated through digital image processing (DIP), response surface methodology (RSM), and analysis of variances (ANOVA). The experimental results show that drilling-induced burr is more severe and surface roughness is worse in BFRP than in CFRP. The composite type influenced the geometrical damages primarily, followed by the feed in the case of burrs and by the cutting speed in the case of surface roughness. The present experimental study suggests that the drilling of BFRP is even more challenging than drilling CFRP from the point of view of burr formation and micro geometrical properties.