  We present a velocimetric and spectropolarimetric analysis of 27 observations
of the 22-Myr M1 star AU Microscopii (Au Mic) collected with the
high-resolution $YJHK$ (0.98-2.35 $\mu$m) spectropolarimeter SPIRou from 2019
September 18 to November 14. Our radial velocity (RV) time-series exhibits
activity-induced fluctuations of 45 m/s RMS, about three times smaller than
those measured in the optical domain, that we filter using Gaussian Process
Regression. We report a 3.9$\sigma$-detection of the recently-discovered 8.46-d
transiting planet AU Mic b, with an estimated mass of $17.1^{+4.7}_{-4.5}$
M$_{\odot}$ and a bulk density of $1.3 \pm 0.4$ g/cm$^{-3}$, inducing a RV
signature of semi-amplitude $K=8.5^{+2.3}_{-2.2}$ m/s in the spectrum of its
host star. A consistent detection is independently obtained when we
simultaneously image stellar surface inhomogeneities and estimate the planet
parameters with Zeeman-Doppler Imaging (ZDI). Using ZDI, we invert the time
series of unpolarized and circularly-polarized spectra into surface brightness
and large-scale magnetic maps. We find a mainly poloidal and axisymmetric field
of 475 G, featuring, in particular, a dipole of 450 G tilted at 19{\deg} to the
rotation axis. Moreover, we detect a strong differential rotation of d$\Omega =
0.167 \pm 0.009$ rad/d shearing the large-scale field, about twice stronger
than that shearing the brightness distribution, suggesting that both
observables probe different layers of the convective zone. Even though we
caution that more RV measurements are needed to accurately pin down the planet
mass, AU Mic b already appears as a prime target for constraining planet
formation models, studying the interactions with the surrounding debris disk,
and characterizing its atmosphere with upcoming space- and ground-based
missions.
