Context. The WASP-10 planetary system is intriguing because different values of radius have been reported for its transiting exoplanet. The host star exhibits activity in terms of photometric variability, which is caused by the rotational modulation of the spots. Moreover, a periodic modulation has been discovered in transit timing of WASP-10 b, which could be a sign of an additional body perturbing the orbital motion of the transiting planet. Aims: We attempt to refine the physical parameters of the system, in particular the planetary radius, which is crucial for studying the internal structure of the transiting planet. We also determine new mid-transit times to confirm or refute observed anomalies in transit timing. Methods: We acquired high-precision light curves for four transits of WASP-10 b in 2010. Assuming various limb-darkening laws, we generated best-fit models and redetermined parameters of the system. The prayer-bead method and Monte Carlo simulations were used to derive error estimates. Results: Three transit light curves exhibit signatures of the occultations of dark spots by the planet during its passage across the stellar disk. The influence of stellar activity on transit depth is taken into account while determining system parameters. The radius of WASP-10 b is found to be no greater than 1.03+0.07-0.03 Jupiter radii, a value significantly smaller than most previous studies indicate. We calculate interior structure models of the planet, assuming a two-layer structure with one homogeneous envelope atop a rock core. The high value of the WASP-10 b's mean density allows one to consider the planet's internal structure including 270 to 450 Earth masses of heavy elements. Our new mid-transit times confirm that transit timing cannot be explained by a constant period if all literature data points are considered. They are consistent with the ephemeris assuming a periodic variation of transit timing. We show that possible starspot features affecting the transit's ingress or egress cannot reproduce variations in transit timing at the observed amplitude. Based on observations collected at the Centro Astronómico Hispano Alemán (CAHA), operated jointly by the Max-Planck Institut für Astronomie and the Instituto de Astrofisica de Andalucia (CSIC).Photometric data are available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/535/A7