The cool white dwarf WD 1856+534 was found to be transited by a Jupiter-sized object with a mass at or below 14 M_Jup. We used the GTC telescope to obtain and analyse the photometry and low-resolution spectroscopy of six transits of WD 1856+534 b, with the intention of deriving the slope of the transmission spectrum. Such a slope, assuming a cloud-free atmosphere dominated by Rayleigh scattering of the particles in its atmosphere, could be translated into an estimation of the mass of WD 1856+534 b. However, the resultant transmission spectrum is essentially flat and therefore permits only the determination of lower mass limits of 2.4 M_Jup at the 2σ level, or 1.6 M_Jup at 3σ. These limits have implications for some of the formation scenarios proposed for the object. We elaborate on the potential effects of clouds and hazes in our estimations, based on previous studies of Jupiter and Titan. In addition, we detected an Hα absorption feature in the combined spectrum of the host white dwarf, which leads to the assignation of a DA classification and allows the derivation of an independent set of atmospheric parameters. Furthermore, the epochs of five transits were measured with sub-second precision, which demonstrates that additional objects more massive than ≈5 M_Jup and with periods longer than O(100) days could be detected through the light-time effect.