  We construct a family of type IIB string backgrounds that are deformations of
$\rm AdS_3 \times S^3 \times T^4$ with a "squashed" $\rm AdS_3 \times S^3$
metric supported by a combination of NSNS and RR fluxes. They have global $\rm
SU(1,1) \times SU(2)$ symmetry, regular curvature, constant dilaton and
preserve 8 supercharges. Upon compactification to 4 dimensions they reduce to
$\mathcal N=2$ supersymmetric $\rm AdS_2 \times S^2$ solutions with electric
and magnetic Maxwell fluxes. These type IIB supergravity solutions can be found
from the undeformed $\rm AdS_3 \times S^3 \times T^4$ background by a
combination of T-dualities and S-duality. In contrast to T-duality, S-duality
transformations of a type IIB supergravity background do not generally preserve
the classical integrability of the corresponding Green-Schwarz superstring
sigma model. Nevertheless, we show that integrability is preserved in the
present case. Indeed, we find that these backgrounds can be obtained, up to
T-dualities, from an integrable inhomogeneous Yang-Baxter deformation (with
unimodular Drinfel'd-Jimbo R-matrix) of the original $\rm AdS_3 \times S^3$
supercoset model.
