Sirius has always attracted a lot of scientific interest, especially after the discovery of a companion white dwarf at the end of the 19th century. Very early on, the existence of a potential third body was put forward to explain some of the observed properties of the system. We present new coronagraphic observations obtained with VLT/SPHERE that explore, for the very first time, the innermost regions of the system down to 0.2 (0.5 AU) from Sirius A. Our observations cover the near-infrared from 0.95 to 2.3 $mu$m and they offer the best on-sky contrast ever reached at these angular separations. After detailing the steps of our SPHERE/IRDIFS data analysis, we present a robust method to derive detection limits for multi-spectral data from high-contrast imagers and spectrographs. In terms of raw performance, we report contrasts of 14.3 mag at 0.2, ~16.3 mag in the 0.4-1.0 range and down to 19 mag at 3.7. In physical units, our observations are sensitive to giant planets down to 11 $M_{Jup}$ at 0.5 AU, 6-7 $M_{Jup}$ in the 1-2 AU range and ~4 $M_{Jup}$ at 10 AU. Despite the exceptional sensitivity of our observations, we do not report the detection of additional companions around Sirius A. Using a Monte Carlo orbital analysis, we show that we can reject, with about 50% probability, the existence of an 8 $M_{Jup}$ planet orbiting at 1 AU. In addition to the results presented in the paper, we provide our SPHERE/IFS data reduction pipeline at http://people.lam.fr/vigan.arthur/ under the MIT license.
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