The authors prove that the following conditions are equivalent for a convex closed subset of~\(\ell^1\): (1)~\(C\) is a compact set; (2)~\(C\) satisfies the fixed point property for Lipschitzian mappings \(T:C\to C\) which are nonexpansive on \(\overline{\mathrm{co}}\,T(C)\); (3)~\(C\) satisfies the fixed point property for cascading nonexpansive mappings; (4)~for every \(L>2\), \(C\) satisfies the fixed point property for uniformly \(L\)-Lipschitzian mappings. This result is extended to some classes of Banach spaces having a boundedly complete Schauder basis, and also to the space of trace class operators on a Hilbert space, with the trace norm.