  We continue the investigation of supersymmetric extensions of field theories
with a non-standard kinetic term (K field theories) resumed recently.
Concretely, for K field theories which allow for kink or compacton solutions in
1+1 dimensions, i.e., for domain walls in a higher-dimensional context, we find
a simple supersymmetric extension such that the boson field still has the kink
solution, and the field equation for the fermion in the kink background is
linear and is solved by the first spatial derivative of the kink, as is the
case in the corresponding standard supersymmetric theories. This supersymmetric
extension, nevertheless, is peculiar in several aspects. The bosonic part of
the supersymmetric Lagrangian is not equal to the original bosonic K field
Lagrangian, but the bosonic field equations coincide. Further, the field
equation for the bosonic field is produced by the variation of the auxiliary
field and vice versa. This observation may be of some independent interest.
Finally, the presence of kink solutions does not lead to a central extension in
the SUSY algebra, in contrast to the standard case.
