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Fast radio bursts (FRBs) are bright, coherent, short-duration radio transients of as-yet unknown extragalactic origin. 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. Here we present the high-time-resolution (down to 1 $μ$s) 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 (EVN). We observe a range of emission timescales spanning three orders of magnitude, with the shortest component width reaching 3-4 $μ$s (below which we are limited by scattering). This is the shortest timescale measured in any FRB, to date. We demonstrate that all four bursts are highly linearly polarised ($\gtrsim 80\%$), show no evidence for significant circular polarisation ($\lesssim 15\%$), and exhibit a constant polarisation position angle (PPA) during and between bursts. On short timescales ($\lesssim 100$ $μ$s), however, there appear to be subtle (few degree) PPA 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.