  Let $G/H$ be a homogeneous space of reductive type with non-compact $H$. The
study of deformations of discontinuous groups for $G/H$ was initiated by
T.~Kobayashi. In this paper, we show that a standard discontinuous group
$\Gamma$ admits a non-standard small deformation as a discontinuous group for
$G/H$ if $\Gamma$ is isomorphic to a surface group of high genus and its
Zariski closure is locally isomorphic to $SL(2,\mathbb{R})$. Furthermore, we
also prove that if $G/H$ is a symmetric space and admits some non virtually
abelian discontinuous groups, then $G$ contains a Zariski-dense discrete
surface subgroup of high genus acting properly discontinuously on $G/H$. As a
key part of our proofs, we show that for a discrete surface subgroup $\Gamma$
of high genus contained in a reductive group $G$, if the Zariski closure of
$\Gamma$ is locally isomorphic to $SL(2,\mathbb{R})$, then $\Gamma$ admits a
small deformation in $G$ whose Zariski closure is a reductive subgroup of the
same real rank as $G$.
