Improving charge mobility in quantum dot (QD) films is important for the performance of photodetectors, solar cells and light-emitting diodes. However, these applications also require preservation of well defined QD electronic states and optical transitions. Here, we present HgTe QD films that show high mobility for charges transported through discrete QD states. A hybrid surface passivation process efficiently eliminates surface states, provides tunable air-stable n and p doping and enables hysteresis-free filling of QD states evidenced by strong conductance modulation. QD films dried at room temperature without any post-treatments exhibit mobility up to μ ~ 8 cm  V  s at a low carrier density of less than one electron per QD, band-like behaviour down to 77 K, and similar drift and Hall mobilities at all temperatures. This unprecedented set of electronic properties raises important questions about the delocalization and hopping mechanisms for transport in QD solids, and introduces opportunities for improving QD technologies.