Aims. We study the resolved stellar populations and derive the star formation history of the Sculptor dwarf irregular galaxy (SDIG), a gas-rich dwarf galaxy member of the NGC 7793 subgroup in the Sculptor group of galaxies. Methods. We constructed a colour−magnitude diagram (CMD) using archival observations from the Hubble Space Telescope/Advanced Camera for Surveys in order to examine the stellar content of SDIG, as well as the spatial distribution of stars selected within different stellar evolutionary phases. We derived the star formation history of SDIG using a maximum-likelihood fit to the CMD. Results. The CMD shows that SDIG contains stars from 10 Myr to several Gyr old, as revealed from the main sequence, blue loop, luminous asymptotic giant branch, and red giant branch stars. The young stars with ages less than ∼250 Myr show a stellar spatial distribution confined to the central regions of SDIG, and additionally the young main sequence stars exhibit an off-centre density peak. The intermediate-age and older stars as traced by the red giant branch stars are more spatially extended. SDIG is dominated by intermediate-age stars with an average age of 6.4 +1.6 −1.4 Gyr. The average metallicity inferred from the CMD modelling is [M/H] ≈ −1.5 dex. SDIG has a star formation history consistent with a constant star formation rate, except for ages younger than ≈200 Myr. The lifetime average star formation rate is 1.3 +0.4 −0.3 × 10 −3 M yr −1 . More recently than 100 Myr, there has been a burst of star formation at a rate ∼2-3 times higher than the average star formation rate. The inferred recent star formation rate from CMD modelling, 2.7(±0.5) × 10 −3 M yr −1 , is higher than inferred from the Hα flux of the galaxy; we interpret this to mean that the upper end of the initial mass function is not being fully sampled due to the low star formation rate. Additionally, an observed lack of bright blue stars in the CMD could indicate a downturn in star formation rate on 10 7 -yr timescales. A previous star formation enhancement appears to have occurred between 600−1100 Myr ago, with amplitude similar to the most recent 100 Myr. Older bursts of similar peak star formation rate and duration would not be resolvable with these data. The observed enhancements in star formation suggest that SDIG is able to sustain a complex star formation history without the effect of gravitational interactions with its nearest massive galaxy. Integrating the star formation rate over the entire history of SDIG yields a total stellar mass equal to 1.77 +0.71 −0.72 × 10 7 M , and a current V-band stellar mass-to-light ratio equal to 3.2 M /L .