Using the scalar field dark matter (SFDM) model, it is proposed that galaxies form by condensation of a scalar field (SF) very early in the Universe, forming Bose–Einstein condensate (BEC) drops (i.e. in this model, the haloes of galaxies are gigantic drops of SF). Here, as in the Λ cold dark matter (LCDM) model, large structures form by hierarchy, and thus all the predictions of the LCDM model at large scales are reproduced by the SFDM model. This model predicts that all galaxies must be very similar and must exist for larger redshifts than in the LCDM model. In this paper, we show that BEC dark matter haloes fit the high-resolution rotation curves of a sample of 13 low-surface-brightness galaxies. We compare our fits to those obtained using Navarro–Frenk–White and pseudo-isothermal (PI) profiles. We have found better agreement with the SFDM and PI profiles. The mean value of the logarithmic inner density slopes is α=−0.27 ± 0.18. As a second result, we find a natural way to define the core radius with the advantage of being model-independent. Using this new definition in the BEC density profile, we find that the recent observation of the constant dark matter central surface density can be reproduced. We conclude that, in light of the difficulties that the standard model is currently facing, the SFDM model could be a worthy alternative to enable us to continue exploring further.