We have used infrared images obtained with the Wide Field Camera 3 on board the Hubble Space Telescope to search for planetary-mass brown dwarfs in the star-forming cluster IC 348. In those images, we have identified 12 objects that have colors indicative of spectral types later than M8, corresponding to masses of ≲ 30 M Jup at the age of IC 348. The four brightest candidates have been observed with spectroscopy, all of which are confirmed to have late types. Two of those candidates appear to be young, and thus are likely members of the cluster, while the ages and membership of the other two candidates are uncertain. One of the former candidates is the faintest known member of IC 348 in extinction-corrected K s and is expected to have a mass of 4-5 M Jup based on evolutionary models and an assumed age of 3 Myr. Four of the remaining eight candidates have ground-based photometry that further supports their candidacy as brown dwarfs, some of which are fainter (and potentially less massive) than the known members.