  HD106906b is an ~11$M_{\mathrm{Jup}}$, ~15Myr old directly-imaged exoplanet
orbiting at an extremely large distance from its host star. The wide separation
(7.11 arcsec) between HD106906b and its host star greatly reduces the
difficulty in direct-imaging observations, making it one of the most favorable
directly-imaged exoplanets for detailed characterization. In this paper, we
present HST/WFC3/IR time-resolved observations of HD106906b in the F127M,
F139M, and F153M bands. We have achieved ~1% precision in the lightcurves in
all three bands. The F127M lightcurve demonstrates marginally-detectable
($2.7\sigma$ significance) variability with a best-fitting period of 4 hr,
while the lightcurves in the other two bands are consistent with flat lines. We
construct primary-subtracted deep images and use these images to exclude
additional companions to HD106906 that are more massive than
4$M_{\mathrm{Jup}}$ and locate at projected distances of more than ~500 au. We
measure the astrometry of HD106906b in two HST/WFC3 epochs and achieve
precisions better than 2.5 mas. The position angle and separation measurements
do not deviate from those in the 2004 HST/ACS/HRC images for more than
$1\sigma$ uncertainty. We provide the HST/WFC3 astrometric results for 25
background stars that can be used as reference sources in future precision
astrometry studies. Our observations also provide the first 1.4-micron water
band photometric measurement for HD106906b. HD106906b's spectral energy
distribution and the best-fitting BT-Settl model have an inconsistency in the
1.4-micron water absorption band, which highlights the challenges in modeling
atmospheres of young planetary-mass objects.
