We present a novel passive satellite remote sensing approach for observing the three-dimensional distribution of aerosols emitted from wildfires. This method, called AEROS5P, retrieves vertical profiles of aerosol extinction from measurements of the TROPOMI satellite sensor onboard the Sentinel 5 Precursor mission. It uses a Tikhonov-Phillips regularization which iteratively fits cloud-free near-infrared and visible selected reflectances to simultaneously adjust the vertical distribution and abundance of aerosols. The information on the altitude of the aerosol layers is provided by TROPOMI measurements of the reflectance spectra at the oxygen A-band near 760 nm. In the present paper, we use this new approach for observing the daily evolution of the three-dimensional distribution of biomass burning aerosols emitted by Australian wildfires on 20-24 December 2019. Aerosol optical depths (AOD) derived by vertical integration of the aerosol extinction profiles retrieved by AEROS5P are compared with MODIS, VIIRS and AERONET coincident observations. They show a good agreement in the horizontal distribution of biomass burning aerosols, with a correlation coefficient of 0.86 and a mean absolute error of 0.2 with respect to VIIRS. A fair agreement is found between coincident transects of vertical profiles of biomass burning aerosols derived from AEROS5P and from the CALIOP spaceborne lidar. The mean altitude of these aerosols derived from these two measurements show a good agreement, with a small mean bias (185 m) and a correlation coefficient of 0.83. Moreover, AEROS5P observations reveal the height of injection of the biomass burning aerosols in 3D. The highest injection heights during the period of analysis are coincident with the largest fire radiative power derived from MODIS. Consistency is also found with respect to the vertical stability of the atmosphere.