Context. Accretion and ejection are tightly connected and represent the fundamental mechanisms regulating star formation. However, the exact physical processes involved are not yet fully understood. Aims. We present high angular and spectral resolution observations of the Brγ emitting region in the Herbig Ae star HD 163296 (MWC 275) in order to probe the origin of this line and constrain the physical processes taking place at sub-AU scales in the circumstellar region. Methods. By means of VLTI-AMBER observations at high spectral resolution (R ∼ 12 000), we studied interferometric visibilities, wavelength-differential phases, and closure phases across the Brγ line of HD 163296. To constrain the physical origin of the Brγ line in Herbig Ae stars, all the interferometric observables were compared with the predictions of a line radiative transfer disc wind model. Results. The measured visibilities clearly increase within the Brγ line, indicating that the Brγ emitting region is more compact than the continuum. By fitting a geometric Gaussian model to the continuum-corrected Brγ visibilities, we derived a compact radius of the Brγ emitting region of ∼0.07 ± 0.02 AU (Gaussian half width at half maximum; or a ring-fit radius of ∼0.08 ± 0.02 AU). To interpret the observations, we developed a magneto-centrifugally driven disc wind model. Our best disc wind model is able to reproduce, within the errors, all the interferometric observables and it predicts a launching region with an outer radius of ∼0.04 AU. However, the intensity distribution of the entire disc wind emitting region extends up to ∼0.16 AU. Conclusions. Our observations, along with a detailed modelling of the Brγ emitting region, suggest that most of the Brγ emission in HD 163296 originates from a disc wind with a launching region that is over five times more compact than previous estimates of the continuum dust rim radius.