We report detailed 75 As-NQR investigations of the locally non-centrosymmetric superconductor SrPtAs. The spin-lattice relaxation studies prove weakly coupled multi-gap superconductivity. A retardation of the decay in 1/T1T evidences a nodeless (fully gapped) superconducting state on the complex multi-pocket Fermi surface, which is consistent with an anisotropic s-wave order parameter and with proposed unconventional f -wave and chiral d -wave symmetries. A quantitative analysis of these models favors the unconventional f -wave state.