Analogues of extreme Local Group galaxies with very low surface brightnesses and large effective radii must exist elsewhere, still hidden due to current detection limits. We report the serendipitous discovery of the low-mass almost-dark galaxy TTT J1237327+143535 in the Virgo Cluster, observed using the Two-meter Twin Telescope at the Teide Observatory. TTT J1237327+143535 has a central surface brightness of 27.9 ± 0.2 mag arcsec-2 in the g' band, an effective radius of 0.9 ± 0.1 kpc, and a total stellar mass of (2.2 ± 0.4)×106 M⊙. Its effective radius and absolute magnitude are similar to those of galaxies And XXI and And XXIII. The discovery of this extremely low-surface-brightness, extended, low-mass galaxy suggests the existence of a significant population of almost-dark galaxies in the Virgo Cluster. An in-depth analysis of the Next Generation Virgo Cluster Survey, as well as upcoming Rubin data releases and the 10-year Rubin Legacy Survey of Space and Time, is expected to reveal large samples of this extreme galaxy population, which will offer insights into galaxy formation in extreme conditions.