  We investigate the theory of the bosonic-fermionic noncommutativity,
$[x^{\mu},\theta^{\alpha}] = i \lambda^{\mu \alpha}$, and the Wess-Zumino model
deformed by the noncommutativity. Such noncommutativity links well-known
space-time noncommutativity to superspace non-anticommutativity. The
deformation has the nilpotency. We can explicitly evaluate noncommutative
effect in terms of new interactions between component fields. The interaction
terms that have Grassmann couplings are induced. The noncommutativity does
completely break full $\mathcal{N}=1$ supersymmetry to $ \mathcal{N} = 0 $
theory in Minkowski signature. Similar to the space-time noncommutativity, this
theory has higher derivative terms and becomes non-local theory. However this
non-locality is milder than the space-time noncommutative field theory. Due to
the nilpotent feature of the coupling constants, we find that there are only
finite number of Feynman diagrams that give noncommutative corrections at each
loop order.
