We present an analysis of deep Hubble Space Telescope (HST)/Wide Field Camera 3 near-IR (NIR) imaging data of the globular cluster (GC) M4. The best-photometry NIR color-magnitude diagram (CMD) clearly shows the main sequence extending toward the expected end of the hydrogen-burning limit and going beyond this point toward fainter sources. The white dwarf (WD) sequence can be identified. As such, this is the deepest NIR CMD of a GC to date. Archival HST optical data were used for proper-motion cleaning of the CMD and for distinguishing the WDs from brown dwarf (BD) candidates. Detection limits in the NIR are around F110W ≈ 26.5 mag and F160W ≈ 27 mag, and in the optical around F775W ≈ 28 mag. Comparing our observed CMDs with theoretical models, we conclude that we have reached beyond the H-burning limit in our NIR CMD and are probably just above or around this limit in our optical-NIR CMDs. Thus, any faint NIR sources that have no optical counterpart are potential BD candidates, since the optical data are not deep enough to detect them. We visually inspected the positions of NIR sources that are fainter than the H-burning limit in F110W and for which the optical photometry did not return a counterpart. We found in total five sources for which we did not get an optical measurement. For four of these five sources, a faint optical counterpart could be visually identified, and an upper optical magnitude was estimated. Based on these upper optical magnitude limits, we conclude that one source is likely a WD, one source could be either a WD or BD candidate, and the remaining two sources agree with being BD candidates. No optical counterpart could be detected for just one source, which makes this source a good BD candidate. We conclude that we found in total four good BD candidates.