  We classify birationally rigid orbifold Fano 3-folds of index one defined by
$5 \times 5$ Pfaffians. We give a sharp criterion for birational rigidity of
these families based on the type of singularities that the varieties admit.
Various conjectures are born out of our study, highlighting a possible approach
to the classification of terminal Fano 3-folds. The birationally rigid cases
are the first known rigid examples of Fanos that are not (weighted) complete
intersection.
