The black hole transient MAXI J1820+070 displayed optical P Cyg profiles and other wind-related emission line features during the hard state of its discovery outburst. We present near-infrared (NIR) spectroscopy covering the different accretion states of the system during this event. Our eight-epoch data set (VLT/X-shooter) reveals strong variability in the properties of the NIR emission lines. This includes absorption troughs and extended emission line wings with kinetic properties that are remarkably similar to those inferred from the wind signatures observed in optical emission lines, indicating that they most likely trace the same accretion disc wind. Unlike the optical features, these NIR signatures are not exclusive of the hard state, as they are also witnessed across the soft state with similar observational properties. This supports the presence of a relatively steady outflow during the entire outburst of the system, and it represents the first detection of an accretion disc wind in a black hole soft state at energies other than X-rays. We discuss the visibility of the wind as a function of the spectral band and the potential of NIR spectroscopy for wind studies, in particular during luminous accretion phases.