Fast radio bursts (FRBs) exhibit a wide variety of spectral, temporal and polarimetric properties, which can unveil clues into their emission physics and propagation effects in the local medium. FRBs are challenging to study at very high time resolution due to the precision needed to constrain the dispersion measure, signal-to-noise limitations, and also scattering from the intervening medium. Here we present the high-time-resolution (down to 1 μs) polarimetric properties of four 1.7-GHz bursts from the repeating FRB 20180916B, which were detected in voltage data during observations with the European VLBI Network. In these bursts we observe a range of emission timescales spanning three orders of magnitude, the shortest component width reaching 3-4 μs (below which we are limited by scattering). We demonstrate that all four bursts are highly linearly polarised (≥ 80%), show no evidence for significant circular polarisation (≤ 15%), and exhibit a constant polarisation position angle during and between bursts. On short timescales (≤ 100 μs), however, there appear to be subtle (few degree) polarisation position angle variations across the burst profiles. These observational results are most naturally explained in an FRB model where the emission is magnetospheric in origin, as opposed to models where the emission originates at larger distances in a relativistic shock.