Plasmonic nanostructures concentrate optical fields into nanoscale volumes, which is useful for plasmonic nanolasers, surface enhanced Raman spectroscopy, and white-light generation. However, the short lifetimes of the emissive plasmons correspond to a rapid depletion of the plasmon energy, preventing further enhancement of local optical fields. Dark (subradiant) plasmons have longer lifetimes, but their resonant wavelengths cannot be tuned over a broad wavelength range without changing the overall geometry of the nanostructures. We have discovered a new type of subradiant plasmon with a narrow (~5 nm) resonant linewidth that can be easily tuned by changing the height of large (> 100 nm) gold nanoparticles arranged in a two-dimensional array. At resonance, strong coupling between out-of-plane nanoparticle dipolar moments suppresses radiative decay, trapping light in the plane of the array and strongly localizing optical fields on each nanoparticle. This new mechanism can open applications for subradiant plasmons because height-controlled nanoparticle arrays can be manufactured over wafer-scale areas on a variety of substrates.