  We consider a baby--Skyrme model with Dzyaloshinskii--Moriya interaction
(DMI) and two types of potential terms. The model has a close connection with
the vacuum functional of fermions coupled with $O(3)$ nonlinear $\bm{n}$-fields
and with a constant $SU(2)$ gauge background. The energy functional is derived
from the heat-kernel expansion for the fermion determinant. The model possesses
normal skyrmions with topological charge $Q = 1$. The restricted version of the
model also includes both the weak-compacton case (at the boundary, not
continuously differentiable) and genuine-compacton case (continuously
differentiable). The model consists of only the Skyrme term, and the DMI
provides soliton solutions that are known as \textit{skyrmions without any
potential}. The BPS equation in the supersymmetric soliton models implies that
the impurity coupling is closely related to the DMI. Therefore, the effect of
an exponentially localized DMI is also studied in the present model.
