Aims. The formation epoch of protostellar disks is debated because of the competing roles of rotation, turbulence, and magnetic fields in the early stages of low-mass star formation. Magnetohydrodynamics simulations of collapsing cores predict that rotationally supported disks may form in strongly magnetized cores through ambipolar diffusion or misalignment between the rotation axis and the magnetic field orientation. Detailed studies of individual sources are needed to cross check the theoretical predictions. Methods. We present 0.06–0.1′′ resolution images at 350 GHz toward B1b-N and B1b-S, which are young class 0 protostars, possibly first hydrostatic cores. The images have been obtained with ALMA, and we compare these data with magnetohydrodynamics simulations of a collapsing turbulent and magnetized core. Results. The submillimeter continuum emission is spatially resolved by ALMA. Compact structures with optically thick 350 GHz emission are detected toward both B1b-N and B1b-S, with 0.2 and 0.35′′ radii (46 and 80 au at the Perseus distance of 230 pc), within a more extended envelope. The flux ratio between the compact structure and the envelope is lower in B1b-N than in B1b-S, in agreement with its earlier evolutionary status. The size and orientation of the compact structure are consistent with 0.2′′ resolution 32 GHz observations obtained with the Very Large Array as a part of the VANDAM survey, suggesting that grains have grown through coagulation. The morphology, temperature, and densities of the compact structures are consistent with those of disks formed in numerical simulations of collapsing cores. Moreover, the properties of B1b-N are consistent with those of a very young protostar, possibly a first hydrostatic core. These observations provide support for the early formation of disks around low-mass protostars.