  We study the Fekete--Szeg\"{o} problem on the open unit ball of a complex
Banach space. Namely, the Fekete--Szeg\"{o} inequalities are proved for the
class of spirallike mappings relative to an arbitrary strongly accretive
operator, and some of its subclasses. Next, we consider families of non-linear
resolvents for holomorphically accretive mappings vanishing at the origin. We
solve the Fekete--Szeg\"{o} problem over these families.
