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$_{rm{Jup}}$. We used the GTC telescope to obtain and analyse photometry and low resolution spectroscopy of six transits of WD 1856+534 b, with the intention to derive the slope of the transmission spectrum, towards an eventual detection of Rayleigh scattering of the particles in its atmosphere. Such a slope, assuming a cloud-free atmosphere dominated by Rayleigh scattering, 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$_{rm{Jup}}$ at the 2-$sigma$ level, or 1.6 M$_{rm{Jup}}$ at 3-$sigma$. These limits have implications for some of the proposed formation scenarios 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 H$alpha$ absorption feature in the combined spectrum of the host white dwarf, that leads to the assignation of a DA classification and allows 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 $approx$5 M$_{rm{Jup}}$ and with periods longer than $O(100)$ days could be detected through the light travel time effect
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