We carried out 12 CO(1−0) and 12 CO(2−1) observations of 21 different regions in the vicinity of M 86, NGC 4438, and along the 120 kpc-long, Hα-emitting filamentary trail that connects them, aiming to test whether molecular gas can survive to be transferred from a spiral to an elliptical galaxy in Virgo's 10 7 K intracluster medium (ICM). We targeted Hα-emitting regions that could be associated with the interface between cold molecular clouds and the hot ionized ICM. The data, obtained with the 30 m telescope of the Institut de Radioastronomie Millimétrique, led to the detection of molecular gas close to M 86. CO gas with a recession velocity that is similar to that of the stars, −265 km s −1 , and with a corresponding H 2 mass of 2 × 10 7 M , was detected ∼10 kpc southeast of the nucleus of M 86, near the peak of its H i emission. We argue that it is possible for this molecular gas either to have formed in situ from H i, or to have been stripped from NGC 4438 directly in molecular form. In situ formation is nonetheless negligible for the 7 × 10 6 M of gas detected at 12:26:15.9+12:58:49, at ∼10 kpc northeast of M 86, where no (strong) H i emission is present. This detection provides evidence for the survival of molecular gas in filaments for timescales of ∼100 Myr. An amount equivalent to 5 × 10 7 M of H 2 gas that could be lost to the ICM or to neighboring galaxies was also discovered in the tidal tail northwest of NGC 4438. A scenario in which gas was alternatively brought to M 86 from NGC 4388 was also examined but it was considered unlikely because of the non-detection of CO below or at the H I stream velocities, 2000-2700 km s −1 .