  We study Yang-Baxter sigma models with deformed 4D Minkowski spacetimes
arising from classical $r$-matrices associated with $\kappa$-deformations of
the Poincar\'e algebra. These classical $\kappa$-Poincar\'e $r$-matrices
describe three kinds of deformations: 1) the standard deformation, 2) the
tachyonic deformation, and 3) the light-cone deformation. For each deformation,
the metric and two-form $B$-field are computed from the associated $r$-matrix.
The first two deformations, related to the modified classical Yang-Baxter
equation, lead to T-duals of dS$_4$ and AdS$_4$\,, respectively. The third
deformation, associated with the homogeneous classical Yang-Baxter equation,
leads to a time-dependent pp-wave background. Finally, we construct a Lax pair
for the generalized $\kappa$-Poincar\'e $r$-matrix that unifies the three kinds
of deformations mentioned above as special cases.
