During 2015, the Relativistic Heavy Ion Collider (RHIC) provided collisions of transversely polarized protons with Au and Al nuclei for the first time, enabling the exploration of transverse-single-spin asymmetries with heavy nuclei. Large single-spin asymmetries in very forward neutron production have been previously observed in transversely polarized p + p collisions at RHIC, and the existing theoretical framework that was successful in describing the single-spin asymmetry in p + p collisions predicts only a moderate atomic-mass-number ( A ) dependence. In contrast, the asymmetries observed at RHIC in p + A collisions showed a surprisingly strong A dependence in inclusive forward neutron production. The observed asymmetry in p + Al collisions is much smaller, while the asymmetry in p + Au collisions is a factor of 3 larger in absolute value and of opposite sign. The interplay of different neutron production mechanisms is discussed as a possible explanation of the observed A dependence.