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.
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