  We study the quantum properties of two theories with a non-anticommutative
(or nilpotent) chiral singlet deformation of N=(1,1) supersymmetry: the abelian
model of a vector gauge multiplet and the model of a gauge multiplet
interacting with a neutral hypermultiplet. In spite of the presence of a
negative-mass-dimension coupling constant (deformation parameter), both
theories are shown to be finite in the sense that the full effective action is
one-loop exact and contains finitely many divergent terms, which vanish
on-shell. The beta-function for the coupling constant is equal to zero. The
divergencies can all be removed off shell by a redefinition of one of the two
scalar fields of the gauge multiplet. These notable quantum properties are
tightly related to the existence of a Seiberg-Witten-type transformation in
both models.
