We consider a novel configuration of plasmonic nanoparticles exhibiting ultra-high sensitivity of the plasmon-polariton resonance to environmental refractive index variations, resulting in a record figure of merit for single-particle sensing. The suggested configuration consists of a metal-coated silica nanorod, whose cylindrical symmetry is broken by a nanosized split resulting in a split-cylinder resonator. Silver and gold nanoparticles are analyzed using two- as well as three-dimensional finite element numerical simulations, revealing that the new nanoparticle configuration facilitates an easy tunability of the resonance and ensures high field enhancement in the split region. Considering single-particle sensing of environmental index variations, we demonstrate that these nanoparticles allow one to achieve record-high figures of merit along with unprecedented spatial resolution.