  We study N=1 supersymmetric gauge theories coupled with higher derivative
chiral models in four dimensions in the off-shell superfield formalism. We
solve the equation of motion for the auxiliary fields and find two distinct
on-shell structures of the Lagrangian that we call the canonical and
non-canonical branches characterized by zero and non-zero auxiliary fields,
respectively. We classify BPS states of the models in Minkowski and Euclidean
spaces. In Minkowski space, we find Abelian and non-Abelian vortices,
vortex-lumps (or gauged lumps with fractional lump charges) as 1/2 BPS states
in the canonical branch and higher derivative generalization of vortices and
vortex-(BPS)baby Skyrmions (or gauged BPS baby Skyrmions with fractional baby
Skyrme charges) as 1/4 BPS states in the non-canonical branch. In
four-dimensional Euclidean space, we find Yang-Mills instantons trapped inside
a non-Abelian vortex, intersecting vortices, intersecting vortex-(BPS)baby
Skyrmions as 1/4 BPS states in the canonical branch but no BPS states in the
non-canonical branch other than those in the Minkowski space.
