It is shown that a uniform algebra can have a nonzero bounded point derivation while having no nontrivial Gleason parts. Conversely, a uniform algebra can have a nontrivial Gleason part while having no nonzero, even possibly unbounded, point derivations.
Ghosh, Swarup and Izzo, Alexander J., "One-point Gleason parts and point derivations in uniform algebras" (2022). Faculty Articles & Research. 3.