We report the growth and characterization of InAs nanowires capped with a 0.5-1 nm epitaxial InP shell. The low-temperature field-effect mobility is increased by a factor 2-5 compared to bare InAs nanowires. We extract the highest low-temperature peak electron mobilities obtained for nanowires to this date, exceeding 20 000 cm2 V s-1. The electron density in the nanowires, determined at zero gate voltage, is reduced by an order of magnitude compared to uncapped InAs nanowires. For smaller diameter nanowires we find an increase in electron density, which can be related to the presence of an accumulation layer at the InAs/InP interface. However, compared to the surface accumulation layer in uncapped InAs, this electron density is much reduced. We suggest that the increase in the observed field-effect mobility can be attributed to an increase of conduction through the inner part of the nanowire and a reduction of the contribution of electrons from the low-mobility accumulation layer. Furthermore the shell around the InAs reduces the surface roughness scattering and ionized impurity scattering in the nanowire.