Searches for planetary transits carried out in open and globular clusters have to date yielded only a handful of weak, unconfirmed candidates. These results have been interpreted as either being insignificant, or evidence that the cluster chemical or dynamical environment inhibits planetary formation or survival. Most campaigns have been limited by small sample statistics or systematics from ground-based photometry. We performed a search for transiting planets and variables in a deep stellar field of NGC 6397 imaged by HST-ACS over 126 orbits. We analyzed 5078 light curves, including a careful selection of 2215 cluster-member M0-M9 dwarfs. The light curves were corrected for systematic trends and inspected using several tools. No high-significance planetary candidate is detected. We compared this null detection with the most recent results from Kepler, showing that no conclusive evidence of lower planet incidence can be drawn. However, a very small photometric jitter is measured for early-M cluster members (≲2 mmag on 98% of them), which may be worth targeting in the near future with more optimized campaigns. Twelve variable stars are reported for the first time. Based on observations with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS 5-26555.