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Transmission spectroscopy is a prime method to study the atmospheres of extrasolar planets. We obtained a high-resolution spectral transit time series of the hot Jupiter WASP-121 b with CRIRES+ to study its atmosphere via transmission spectroscopy of the He I λ10833 triplet lines. Our analysis shows a prominent He I λ10833 absorption feature moving along with the planetary orbital motion, which shows an observed, transit-averaged equivalent width of approximately 30 mÅ, a slight redshift, and a depth of about 2%, which can only be explained by an atmosphere overflowing its Roche lobe. We carried out 3D hydrodynamic modeling to reproduce the observations, which favors asymmetric mass loss with a more pronounced leading tidal tail, possibly also explaining observational evidence for additional absorption stationary in the stellar rest frame. A trailing tail is not detectable. From our modeling, we derived estimates of ≥2 × 10¹³ g s⁻¹ for the stellar and 5.4 × 10¹² g s⁻¹ for the planetary mass loss rate, which is consistent with X-ray and extreme-ultraviolet (XUV) driven mass loss in WASP-121 b.