We present ultraviolet (UV) observations of six nearby Type Ia supernovae (SNe Ia) obtained with the Hubble Space Telescope, three of which were also observed in the near-IR (NIR) with Wide-Field Camera 3. UV observations with the Swift satellite, as well as ground-based optical and NIR data provide complementary information. The combined data set covers the wavelength range 0.2–2 μm. By also including archival data of SN 2014J, we analyse a sample spanning observed colour excesses up to E(B − V) = 1.4 mag. We study the wavelength-dependent extinction of each individual SN and find a diversity of reddening laws when characterized by the total-to-selective extinction R V . In particular, we note that for the two SNe with E(B − V) ≳ 1 mag, for which the colour excess is dominated by dust extinction, we find R V  = 1.4 ± 0.1 and R V  = 2.8 ± 0.1. Adding UV photometry reduces the uncertainty of fitted R V by ∼50 per cent allowing us to also measure R V of individual low-extinction objects which point to a similar diversity, currently not accounted for in the analyses when SNe Ia are used for studying the expansion history of the Universe.