Bent Cu-Al-Ni nanopillars (diameters 90-750 nm) show a shape memory effect, SME, for diameters D > 300 nm. The SME and the associated twinning are located in a small deformed section of the nanopillar. Thick nanopillars (D > 300 nm) transform to austenite under heating, including the deformed region. Thin nanopillars (D < 130 nm) do not twin but generate highly disordered sequences of stacking faults in the deformed region. No SME occurs and heating converts only the undeformed regions into austenite. The defect-rich, deformed region remains in the martensite phase even after prolonged heating in the stability field of austenite. A complex mixture of twins and stacking faults was found for diameters 130 nm < D < 300 nm. The size effect of the SME in Cu-Al-Ni nanopillars consists of an approximately linear reduction of the SME between 300 and 130 nm when the SME completely vanishes for smaller diameters.