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Recent observations have reignited interest in a population of dwarf galaxies that are large and diffuse for their mass, often called ultra-diffuse galaxies (UDGs). 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 (UPGs), defined as being 1.5 σ 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 UPGs 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 UPGs associated with redder hosts, as well as those that are closer to the host in projection. In contrast, the UDGs are overall redder and more quiescent compared with normal satellites. We show that the classic definition of UDGs is heavily weighted toward quenched galaxies and thus cannot be used for a study of quenching of mass–size outliers. [ABSTRACT FROM AUTHOR]