Gravitational microlensing is a powerful tool for probing the inner structure of distant quasars. In this context, we have obtained spectropolarimetric observations of the two images of the broad absorption line (BAL) quasar SDSS J081830.46+060138.0 (J0818+0601) at redshift z~~2.35. We first show that J0818+0601 is actually gravitationally lensed, and not a binary quasar. A strong absorption system detected at z=1.0065+/-0.0002 is possibly due to the lensing galaxy. Microlensing is observed in one image and it magnifies the emission lines, the continuum, and the BALs differently. By disentangling the part of the spectrum that is microlensed from the part that is not microlensed, we unveil two sources of continuum that must be spatially separated: a compact one, which is microlensed, and an extended one, which is not microlensed and contributes to two thirds of the total continuum emission. J0818+0601 is the second BAL quasar in which an extended source of rest-frame ultraviolet continuum is found. We also find that the images are differently polarized, suggesting that the two continua might be differently polarized. Our analysis provides constraints on the BAL flow. In particular, we find that the outflow is seen with a nonzero onset velocity, and stratified according to ionization.