We obtained FLAMES GIRAFFE+UVES spectra for both first- and second-generation red giant branch (RGB) stars in the globular cluster (GC) NGC 362 and used them to derive abundances of 21 atomic species for a sample of 92 stars. The surveyed elements include proton-capture (O, Na, Mg, Al, Si), α-capture (Ca, Ti), Fe-peak (Sc, V, Mn, Co, Ni, Cu), and neutron-capture elements (Y, Zr, Ba, La, Ce, Nd, Eu, Dy). The analysis is fully consistent with that presented for twenty GCs in previous papers of this series. Stars in NGC 362 seem to be clustered into two discrete groups along the Na-O anti-correlation with a gap at [O/Na] ~ 0 dex. Na-rich, second generation stars show a trend to be more centrally concentrated, although the level of confidence is not very high. When compared to the classical second-parameter twin NGC 288 with similar metallicity but different horizontal branch type and a much lower total mass, the proton-capture processing in stars of NGC 362 seems to be more extreme, confirming previous analysis. We discovered the presence of a secondary RGB sequence, which is redder than the bulk of the RGB. A preliminary estimate shows that this sequence comprises about 6% of RGB stars. Our spectroscopic data and literature photometry indicate that this sequence is populated almost exclusively by giants rich in Ba and is probably rich in all s-process elements, as found in other clusters. In this regard, NGC 362 joins previously studied GCs like NGC 1851, NGC 6656 (M 22), and NGC 7089 (M 2).