This paper presents data from a 21 cm H I emission drift scan observation of a field partially covering the NGC 4636 galaxy group with the Five-hundred meter Aperture Radio Telescope (FAST). We construct a pipeline to reduce the data, and use SoFiA for source finding. When not contaminated by Radio Frequency Interference (RFI), the FAST observations are capable of detecting all of the galaxies previously detected by the Arecibo Legacy Fast ALFA (ALFALFA) survey in the same region. Comparing to ALFALFA for the detections in common, the FAST data show consistent integrated spectra when the H I disks are spatially unresolved, and capture more flux when the H I disks are resolved. The FAST data further reveal 10 new detections in the region mutually covered with ALFALFA, and 18 new detections beyond the footprint of ALFALFA. All of the new detections have the matching optical counterparts. For the member galaxies of the NGC 4636 group, the detection limit of FAST is deeper by 0.4 dex in H I mass than that of the ALFALFA data. After correcting for the incompleteness caused by RFI contamination, we show that the H I detection rate of galaxies rises steeply with radius out to the virial radius of the group, and flattens beyond that. We also examine four spatially resolved galaxy systems with potential tidal interaction features in detail. Considering that the data have been taken during the "shared-risk" period before a major source of local RFI was eliminated, the results highlight the power of FAST in detecting extragalactic H I.