  We introduce a new elliptic integrable $\sigma$-model in the form of a
two-parameter deformation of the Principal Chiral Model on the group
$\text{SL}_{\mathbb{R}}(N)$, generalising a construction of Cherednik for $N=2$
(up to reality conditions). We exhibit the Lax connection and
$\mathcal{R}$-matrix of this theory, which depend meromorphically on a spectral
parameter valued in the torus. Furthermore, we explain the origin of this model
from an equivariant semi-holomorphic 4-dimensional Chern-Simons theory on the
torus. This approach opens the way for the construction of a large class of
elliptic integrable $\sigma$-models, with the deformed Principal Chiral Model
as the simplest example.
