Sirius has always attracted a lot of scientific interest, especially after the discovery of a companion white dwarf at the end of the 19 th 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 µm 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.