Currently, the use of sustainable products and technologies is growing; consequently, mineral-origin basalt fibre-reinforced polymer (BFRP) composites are becoming more popular in industries. Although BFRP parts require mechanical drilling operations for manufacturing holes for assembly, many challenges make the drilling process difficult. Considering that the cutting force is one of the main parameters characterising the drilling process, this study aims to analyse the influence of feed (mm/rev) and cutting speed (m/min) on the thrust force and model the thrust force in the drilling of BFRP composites through response surface methodology (RSM) and advanced statistical modelling methods. In order to determine main and interaction effects and to calculate the regression coefficients and model parameters, mechanical drilling experiments were performed, and the thrust force was recorded. The raw force data were processed using fast Fourier transformation-based low-pass filtering, and then the calculated thrust force parameters were evaluated relative to various feeds and cutting speeds. In addition, results were compared with those of carbon fibre-reinforced polymer composites. The results of the validation experiments show that both RSM and advanced statistical models accurately predict the thrust force in BFRPs of 96.74% and 95.01%, respectively. However, the advanced statistical model can describe not only the maximum values of the force but also its characteristics at a coefficient of determination of 0.68.