  We present extremely deep Hubble Space Telescope (HST) Wide Field Camera 3
(WFC3) observations of the MUSE Ultra Deep Field (MUDF). This unique region of
the sky contains two quasars at $z \approx$ 3.22 that are separated by only
$\sim$500 kpc, providing a stereoscopic view of gas and galaxies in emission
and absorption across $\sim$10 billion years of cosmic time. We have obtained
90 orbits of HST WFC3 G141 near-infrared grism spectroscopy of this field in a
single pointing, as well as 142 hours of optical spectroscopy with the Very
Large Telescope (VLT) Multi Unit Spectroscopic Explorer (MUSE). The WFC3
(F140W, F125W, and F336W) and archival WFPC2 (F702W and F450W) imaging provides
five-filter photometry that we use to detect 3,375 sources between $z \approx$
0 - 6, including 1,536 objects in a deep central pointing with both
spectroscopic and photometric coverage. The F140W and F336W mosaics reach
exceptional depths of $m_\mathrm{AB}\approx$ 28 and 29, respectively, providing
near-infrared and rest-frame ultraviolet information for 1,580 sources, and we
reach 5$\sigma$ continuum detections for objects as faint as
$m_\mathrm{AB}\approx$ 27 in the grism spectra. The extensive wavelength
coverage of MUSE and WFC3 allows us to measure spectroscopic redshifts for 419
sources, down to galaxy stellar masses of log(M/M$_{\odot}$) $\approx$ 7 at $z
\approx$ 1 - 2. In this publication, we provide the calibrated HST data and
source catalogs as High Level Science Products for use by the community, which
includes photometry, morphology, and redshift measurements that enable a
variety of studies aimed at advancing our models of galaxy formation and
evolution in different environments.
