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We report 31P-NMR and specific-heat measurements on an iron (Fe) based superconductor SrFe2(As0.65P0.35)2 with Tc = 26 K, which have revealed the development of antiferromagnetic correlations in the normal state and unconventional superconductivity (SC) with the nodal gap dominated by the gapless low-lying quasiparticle excitations. The results are consistently argued with an unconventional multiband SC state with the gap-size ratio of different bands being significantly large; the large full gaps in the s±-wave state keep Tc high, whereas a small gap with a nodal structure causes a gapless feature under magnetic field. The present results will help develop insight into the strong material dependence of the SC gap structure in Fe-based superconductors. [ABSTRACT FROM AUTHOR]