  Aims. The initial stages of planet formation may start concurrently with the
formation of a gas-dust protoplanetary disk. This makes the study of the
earliest stages of protoplanetary disk formation crucially important. Here we
focus on dust growth and pebble formation in a protoplanetary disk that is
still accreting from a parental cloud core. Methods. We have developed an
original three-dimensional numerical hydrodynamics code, which computes the
collapse of rotating clouds and disk formation on nested meshes using a novel
hybrid Coarray Fortran-OpenMP approach for distributed and shared memory
parallelization. Dust dynamics and growth are also included in the simulations.
Results. We found that the dust growth from $\sim 1~\mu$m to 1-10~mm already
occurs in the initial few thousand years of disk evolution but the Stokes
number hardly exceeds 0.1 because of higher disk densities and temperatures
compared to the minimum mass Solar nebular. The ratio of the dust-to-gas
vertical scale heights remains rather modest, 0.2--0.5, which may be explained
by the perturbing action of spiral arms that develop in the disk soon after its
formation. The dust-to-gas mass ratio in the disk midplane is highly
nonhomogeneous throughout the disk extent and is in general enhanced by a
factor of several compared to the fiducial 1:100 value. Low St hinders strong
dust accumulation in the spiral arms compared to the rest of the disk and the
nonsteady nature of the spirals is also an obstacle. The spatial distribution
of pebbles in the disk midplane exhibits a highly nonhomogeneous and patchy
character. The total mass of pebbles in the disk increases with time and
reaches a few tens of Earth masses after a few tens of thousand years of disk
evolution. Abridged.
