  We formulate four-dimensional $\mathcal{N} = 1$ supersymmetric nonlinear
sigma models on Hermitian symmetric spaces with higher derivative terms, free
from the auxiliary field problem and the Ostrogradski's ghosts, as gauged
linear sigma models. We then study Bogomol'nyi-Prasad-Sommerfield equations
preserving 1/2 or 1/4 supersymmetries. We find that there are distinct
branches, that we call canonical ($F=0$) and non-canonical ($F\neq 0$)
branches, associated with solutions to auxiliary fields $F$ in chiral
multiplets. For the ${\mathbb C}P^N$ model, we obtain a supersymmetric
${\mathbb C}P^N$ Skyrme-Faddeev model in the canonical branch while in the
non-canonical branch the Lagrangian consists of solely the ${\mathbb C}P^N$
Skyrme-Faddeev term without a canonical kinetic term. These structures can be
extended to the Grassmann manifold $G_{M,N} = SU(M)/[SU(M-N)\times SU(N) \times
U(1)]$. For other Hermitian symmetric spaces such as the quadric surface
$Q^{N-2}=SO(N)/[SO(N-2) \times U(1)])$, we impose F-term (holomorphic)
constraints for embedding them into ${\mathbb C}P^{N-1}$ or Grassmann manifold.
We find that these constraints are consistent in the canonical branch but yield
additional constraints on the dynamical fields thus reducing the target spaces
in the non-canonical branch.
