We present the first analysis of JWST near-infrared spectroscopy of stellar flares from TRAPPIST-1 during transits of rocky exoplanets. Four flares were observed from 0.6–2.8 μm with the Near Infrared Imager and Slitless Spectrograph and 0.6–3.5 μm with the Near Infrared Spectrograph during transits of TRAPPIST-1b, f, and g. We discover Pα and Brβ line emission and characterize flare continuum at wavelengths from 1–3.5 μm for the first time. Observed lines include Hα, Pα–Pϵ, Brβ, He i
λ0.7062 μm, two Ca ii infrared triplet (IRT) lines, and the He i IRT. We observe a reversed Paschen decrement from Pα–Pγ alongside changes in the light-curve shapes of these lines. The continuum of all four flares is well described by blackbody emission with an effective temperature below 5300 K, lower than the temperatures typically observed at optical wavelengths. The 0.6–1 μm spectra were convolved with the Transiting Exoplanet Survey Satellite (TESS) response, enabling us to measure the flare rate of TRAPPIST-1 in the TESS bandpass. We find flares of 1030 erg, large enough to impact transit spectra occur at a rate of
3.6
−
1.3
+
2.1
flare day−1, ∼10× higher than previous predictions from K2. We measure the amount of flare contamination at 2 μm for the TRAPPIST-1b and f transits to be 500 ± 450 and 2100 ± 400 ppm, respectively. We find up to 80% of flare contamination can be removed, with mitigation most effective from 1.0–2.4 μm. These results suggest transits affected by flares may still be useful for atmospheric characterization efforts.