Polluted white dwarfs that have accreted planetary material provide a unique opportunity to probe the geology of exoplanetary systems. However, the nature of the bodies which pollute white dwarfs is not well understood: are they small asteroids, minor planets, or even terrestrial planets? We present a novel method to infer pollutant masses from detections of Ni, Cr and Si. During core--mantle differentiation, these elements exhibit variable preference for metal and silicate at different pressures (i.e., object masses), affecting their abundances in the core and mantle. We model core--mantle differentiation self-consistently using data from metal--silicate partitioning experiments. We place statistical constraints on the differentiation pressures, and hence masses, of bodies which pollute white dwarfs by incorporating this calculation into a Bayesian framework. We show that Ni observations are best suited to constraining pressure when pollution is mantle-like, while Cr and Si are better for core-like pollution. We find 3 systems (WD0449-259, WD1350-162 and WD2105-820) whose abundances are best explained by the accretion of fragments of small parent bodies (). For 2 systems (GD61 and WD0446-255), the best model suggests the accretion of fragments of Earth-sized bodies, although the observed abundances remain consistent () with the accretion of undifferentiated material. This suggests that polluted white dwarfs potentially accrete planetary bodies of a range of masses. However, our results are subject to inevitable degeneracies and limitations given current data. To constrain pressure more confidently, we require serendipitous observation of (nearly) pure core and/or mantle material.