  We study a non-anticommutative chiral non-singlet deformation of the N=(1,1)
abelian gauge multiplet in Euclidean harmonic superspace with a product ansatz
for the deformation matrix, C^{(\alpha\beta)}_{(ik)} =
c^{(\alpha\beta)}b_{(ik)}. This allows us to obtain in closed form the gauge
transformations and the unbroken N=(1,0) supersymmetry transformations
preserving the Wess-Zumino gauge, as well as the bosonic sector of the N=(1,0)
invariant action. As in the case of a singlet deformation, the bosonic action
can be cast in a form where it differs from the free action merely by a scalar
factor. The latter is now given by \cosh^2 (2\bar\phi\sqrt{c^2 b^2}}), with
\bar\phi being one of two scalar fields of the N=(1,1) vector multiplet. We
compare our results with previous studies of non-singlet deformations,
including the degenerate case b^2=0 which preserves the N=(1,1/2) fraction of
N=(1,1) supersymmetry.
