  Heavy neutral leptons (HNLs) with masses around the electroweak scale are
expected to be rather long-lived particles, as a result of the observed
smallness of the active neutrino masses. In this work, we study long-lived HNLs
in $N_R$SMEFT, a Standard Model (SM) extension with singlet fermions to which
we add non-renormalizable operators up to dimension-6. Operators which contain
two HNLs can lead to a sizable enhancement of the production cross sections,
compared to the minimal case where HNLs are produced only via their mixing with
the SM neutrinos. We calculate the expected sensitivities for the ATLAS
detector and the future far-detector experiments: AL3X, ANUBIS, CODEX-b, FASER,
MATHUSLA, and MoEDAL-MAPP in this setup. The sensitive ranges of the HNL mass
and of the active-heavy mixing angle are much larger than those in the minimal
case. We study both, Dirac and Majorana, HNLs and discuss how the two cases
actually differ phenomenologically, for HNL masses above roughly 100 GeV.
