Gaia stellar measurements are currently revolutionizing our knowledge of the evolutionary history of the Milky Way. 3D maps of the interstellar dust provide complementary information and are a tool for a wide range of uses. We built 3D maps of the dust in the Local arm and surrounding regions. To do so, Gaia DR2 photometric data were combined with 2MASS measurements to derive extinction toward stars that possess accurate photometry and relative uncertainties on DR2 parallaxes smaller than 20%. We applied a new hierarchical inversion algorithm to the individual extinctions that is adapted to large datasets and to an inhomogeneous target distribution. Each step associates regularized Bayesian inversions in all radial directions and a subsequent inversion in 3D of all their results. Each inverted distribution serves as a prior for the subsequent step, and the spatial resolution is progressively increased. We present the resulting 3D distribution of the dust in a 6 × 6 × 0.8 kpc³ volume around the Sun. Its main features are found to be elongated along different directions that vary from below to above the mid-plane. The outer part of Carina-Sagittarius, mainly located above the mid-plane, the Local arm/Cygnus Rift around and above the mid-plane, and the fragmented Perseus arm are oriented close to the direction of circular motion. The spur of more than 2 kpc length (nicknamed the split) that extends between the Local Arm and Carina-Sagittarius, the compact near side of Carina-Sagittarius, and the Cygnus Rift below the Plane are oriented along l ~40 to 55°. Dust density images in vertical planes reveal a wavy pattern in some regions and show that the solar neighborhood within ~500 pc remains atypical by its extent above and below the Plane. We show several comparisons with the locations of molecular clouds, HII regions, O stars, and masers. The link between the dust concentration and these tracers is markedly different from one region to the other.