Previous attempts to separate Small Magellanic Cloud (SMC) stars from the Milky Way (MW) foreground stars are based only on the proper motions of the stars. In this paper, we aim to develop a statistical classification technique to effectively separate the SMC stars from the MW stars using a wider set of Gaia data. We aim to reduce the possible contamination from MW stars compared to previous strategies. The new strategy is based on a neural network classifier, applied to the bulk of the Gaia DR3 data. We produce three samples of stars flagged as SMC members, with varying levels of completeness and purity, obtained by application of this classifier. Using different test samples, we validated these classification results and compared them with the results of the selection technique employed in the Gaia Collaboration papers, which was based solely on the proper motions. The contamination of the MW in each of the three SMC samples is estimated to be in the 10-40% range; the "best case" in this range is obtained for bright stars (G<16), which belong to the V los sub-samples, and the "worst case" for the full SMC sample determined by using very stringent criteria based on StarHorse distances. A further check based on the comparison with a nearby area with uniform sky density indicates that the global contamination in our samples is probably close to the low end of the range, around 10%. We provide three selections of SMC star samples with different degrees of purity and completeness, for which we estimate a low contamination level and which we have successfully validated using SMC RR Lyrae, SMC Cepheids, and SMC-MW StarHorse samples.