We present Gemini Multi Object Spectrograph integral-field unit (GMOS-IFU) data of eight compact, low-mass early-type galaxies (ETGs) in the Virgo cluster. We analyze their stellar kinematics and stellar population and present two-dimensional maps of these properties covering the central 5″ 7″ region. We find a large variety of kinematics, from nonrotating to highly rotating objects, often associated with underlying disky isophotes revealed by deep images from the Next Generation Virgo Cluster Survey. In half of our objects, we find a centrally concentrated younger and more metal-rich stellar population. We analyze the specific stellar angular momentum through the λR parameter and find six fast rotators and two slow rotators, one having a thin counterrotating disk. We compare the local galaxy density and stellar populations of our objects with those of 39 more extended low-mass Virgo ETGs from the SMAKCED survey and 260 massive (M > 1010 ) ETGs from the ATLAS3D sample. The compact low-mass ETGs in our sample are located in high-density regions, often close to a massive galaxy, and have, on average, older and more metal-rich stellar populations than less compact low-mass galaxies. We find that the stellar population parameters follow lines of constant velocity dispersion in the mass-size plane, smoothly extending the comparable trends found for massive ETGs. Our study supports a scenario where low-mass compact ETGs have experienced long-lived interactions with their environment, including ram-pressure stripping and gravitational tidal forces, that may be responsible for their compact nature.