A sample of 3984 candidates of the K+ -> pi(+)gamma gamma decay, with an estimated background of 291 +/- 14 events, was collected by the NA62 experiment at CERN during 2017-2018. In order to describe the observed di-photon mass spectrum, the next-to-leading order contribution in chiral perturbation theory was found to be necessary. The decay branching ratio in the full kinematic range is measured to be (9.61 +/- 0.17) x 10(-7). The first search for production and prompt decay of an axion-like particle with gluon coupling in the process K+ -> pi(+) a, a -> gamma gamma is also reported.