Context. Observing the neutral hydrogen distribution across the Universe via redshifted 21 cm line intensity mapping constitutes a powerful probe for cosmology. However, the redshifted 21 cm signal is obscured by the foreground emission from our Galaxy and other extragalactic foregrounds. This paper addresses the capabilities of the BINGO survey to separate such signals. Aims. We show that the BINGO instrumental, optical, and simulations setup is suitable for component separation, and that we have the appropriate tools to understand and control foreground residuals. Specifically, this paper looks in detail at the different residuals left over by foreground components, shows that a noise-corrected spectrum is unbiased, and shows that we understand the remaining systematic residuals by analyzing nonzero contributions to the three-point function. Methods. We use the generalized needlet internal linear combination, which we apply to sky simulations of the BINGO experiment for each redshift bin of the survey. We use binned estimates of the bispectrum of the maps to assess foreground residuals left over after component separation in the final map. Results. We present our recovery of the redshifted 21 cm signal from sky simulations of the BINGO experiment, including foreground components. We test the recovery of the 21 cm signal through the angular power spectrum at different redshifts, as well as the recovery of its non-Gaussian distribution through a bispectrum analysis. We find that non-Gaussianities from the original foreground maps can be removed down to, at least, the noise limit of the BINGO survey with such techniques. Conclusions. Our component separation methodology allows us to subtract the foreground contamination in the BINGO channels down to levels below the cosmological signal and the noise, and to reconstruct the 21 cm power spectrum for different redshift bins without significant loss at multipoles 20≲ l ≲ 500. Our bispectrum analysis yields strong tests of the level of the residual foreground contamination in the recovered 21 cm signal, thereby allowing us to both optimize and validate our component separation analysis