We investigate the sensitivity of the FASER$ u$ detector to new physics in the form of non-standard neutrino interactions. FASER$ u$, which has recently been installed 480 m downstream of the ATLAS interaction point, will for the first time study interactions of multi-TeV neutrinos from a controlled source. Our formalism -- which is applicable to any current and future neutrino experiment -- is based on the Standard Model Effective Theory~(SMEFT) and its counterpart, Weak Effective Field Theory~(WEFT), below the electroweak scale. Starting from the WEFT Lagrangian, we compute the coefficients that modify neutrino production in meson decays and detection via deep-inelastic scattering, and we express the new physics effects in terms of modified flavor transition probabilities. For some coupling structures, we find that FASER$ u$ will be able to constrain interactions that are two to three orders of magnitude weaker than Standard Model weak interactions, implying that the experiment will be indirectly probing new physics at the multi-TeV scale. In some cases, FASER$ u$ constraints will become comparable to existing limits - some of them derived for the first time in this paper - already with $150~$fb${}^{-1}$ of data.
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