  We report the discovery of significant mass/light offsets in the strong
gravitational lensing system SDSS\,J1011$+$0143. We use the high-resolution
\textsl{Hubble Space Telescope} (\textsl{HST}) F555W- and F814W-band imaging
and Sloan Digital Sky Survey (SDSS) spectroscopy of this system, which consists
of a close galaxy pair with a projected separation of $\approx 4.2$ kpc at
$z_{\rm lens} \sim 0.331$ lensing a Ly$\alpha$ emitter (LAE) at $z_{\rm source}
= 2.701$. Comparisons between the mass peaks inferred from lens models and
light peaks from \textsl{HST} imaging data reveal significant spatial
mass/light offsets as large as $1.72 \pm 0.24 \pm 0.34$ kpc in both filter
bands. Such large mass/light offsets, not seen in isolated field lens galaxies
and relaxed galaxy groups, may be related to the interactions between the two
lens galaxies. The detected mass/light offsets can potentially serve as an
important test for the self-interacting dark matter model. However, other
mechanisms such as dynamical friction on spatially differently distributed dark
matter and stars could produce similar offsets. Detailed hydrodynamical
simulations of galaxy-galaxy interactions with self-interacting dark matter
could accurately quantify the effects of different mechanisms. The background
LAE is found to contain three distinct star-forming knots with characteristic
sizes from 116 pc to 438 pc. It highlights the power of strong gravitational
lensing in probing the otherwise too faint and unresolved structures of
distance objects below subkiloparsec or even 100 pc scales through its
magnification effect.
