Transverse single-spin asymmetries of very forward neutral pions generated in polarized p+p collisions allow us to understand the production mechanism in terms of perturbative and nonperturbative strong interactions. During 2017, the RHICf Collaboration installed an electromagnetic calorimeter in the zero-degree region of the STAR detector at the Relativistic Heavy Ion Collider (RHIC) and measured neutral pions produced at pseudorapidity larger than 6 in polarized p+p collisions at sqrt[s]=510 GeV. The large nonzero asymmetries increasing both in longitudinal momentum fraction x_{F} and transverse momentum p_{T} have been observed at low transverse momentum p_{T}<1 GeV/c for the first time, at this collision energy. The asymmetries show an approximate x_{F} scaling in the p_{T} region where nonperturbative processes are expected to dominate. A non-negligible contribution from soft processes may be necessary to explain the nonzero neutral pion asymmetries.