  Recent observations have reignited interest in a population of dwarf galaxies
that are large and diffuse for their mass, often called ultra-diffuse galaxies.
However, the origin and evolution of these mass-size outliers and the role of
the environment are still unclear. Using the exquisitely deep and wide Hyper
Suprime-Cam Strategic Survey images, we search for ultra-puffy galaxies,
defined as being $1.5\sigma$ larger than the average size for their mass,
around Milky Way-like galaxies. We present the sizes and radial distributions
of mass-size outliers and derive their quenched fraction to explore the impact
of the environment. Surprisingly, despite being outliers in size, the
ultra-puffy galaxies have a similar quenched fraction as normal-sized
satellites of Milky Way analogs in both observations and simulations,
suggesting that quenching is not tied to being a mass-size outlier. The
quenched fraction is higher for the ultra-puffy galaxies associated with redder
hosts, as well as those that are closer to the host in projection. In contrast,
the ultra-diffuse galaxies are overall redder and more quiescent compared with
normal satellites. We show that the classic definition of ultra-diffuse
galaxies is heavily weighted toward quenched galaxies and thus cannot be used
for a study of quenching of mass-size outliers.
