Context. On the asymptotic giant branch (AGB) low- and intermediate-mass stars eject a large fraction of their envelope, but the mechanism driving these outflows is still poorly understood. For oxygen-rich AGB stars, the wind is thought to be driven by radiation pressure caused by scattering of radiation off dust grains. Aims: We study the photosphere, the warm molecular layer, and the inner wind of the close-by oxygen-rich AGB star R Doradus. We focus on investigating the spatial distribution of the dust grains that scatter light and whether these grains can be responsible for driving the outflow of this star. Methods: We use high-angular-resolution images obtained with SPHERE/ZIMPOL to study R Dor and its inner envelope in a novel way. We present observations in filters V, cntHα, and cnt820 and investigate the surface brightness distribution of the star and of the polarised light produced in the inner envelope. Thanks to second-epoch observations in cntHα, we are able to see variability on the stellar photosphere. We study the polarised-light data using a continuum-radiative-transfer code that accounts for direction-dependent scattering of photons off dust grains. Results: We find that in the first epoch the surface brightness of R Dor is asymmetric in V and cntHα, the filters where molecular opacity is stronger, while in cnt820 the surface brightness is closer to being axisymmetric. The second-epoch observations in cntHα show that the morphology of R Dor has changed completely in a timespan of 48 days to a more axisymmetric and compact configuration. This variable morphology is probably linked to changes in the opacity provided by TiO molecules in the extended atmosphere. The observations show polarised light coming from a region around the central star. The inner radius of the region from where polarised light is seen varies only by a small amount with azimuth. The value of the polarised intensity, however, varies by between a factor of 2.3 and 3.7 with azimuth for the different images. We fit the radial profile of the polarised intensity using a spherically symmetric model and a parametric description of the dust density profile, ρ(r) = ρ°r- n. On average, we find exponents of - 4.5 ± 0.5 that correspond to a much steeper density profile than that of a wind expanding at constant velocity. The dust densities we derive imply an upper limit for the dust-to-gas ratio of ~2 × 10-4 at 5.0 R⋆. Considering all the uncertainties in observations and models, this value is consistent with the minimum values required by wind-driving models for the onset of a wind, of ~3.3 × 10-4. However, if the steep density profile we find extends to larger distances from the star, the dust-to-gas ratio will quickly become too small for the wind of R Dor to be driven by the grains that produce the scattered light.