  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.
