We investigate how SL(2,Z) duality is realized in nonrelativistic type IIB superstring theory, which is a self-contained corner of relativistic string theory. Within this corner, we realize manifestly SL(2,Z)-invariant (p, q)-string actions. The construction of these actions imposes a branching between strings of opposite charges associated with the two-form fields. The branch point is determined by these charges and the axion background field. Both branches must be incorporated in order to realize the full SL(2,Z) group. Besides these string actions, we also construct D-instanton and D3-brane actions that manifestly realize the branched SL(2,Z) symmetry.