We present photometry and spectroscopy of the peculiar Type II supernova (SN) SN 2010jp, also named PTF10aaxi. The light curve exhibits a linear decline with a relatively low peak absolute magnitude of only −15.9 (unfiltered), and a low radioactive decay luminosity at late times, which suggests a low synthesized nickel mass of M (56 Ni) ≲ 0.003  M⊙. Spectra of SN 2010jp display an unprecedented triple-peaked Hα line profile, showing (1) a narrow (full width at half-maximum >rsim800 km s−1) central component that suggests shock interaction with dense circumstellar material (CSM); (2) high-velocity blue and red emission features centred at −12 600 and +15 400 km s−1, respectively; and (3) very broad wings extending from −22 000 to +25 000 km s−1. These features persist over multiple epochs during the ∼100 d after explosion. We propose that this line profile indicates a bipolar jet-driven explosion, with the central component produced by normal SN ejecta and CSM interaction at mid and low latitudes, while the high-velocity bumps and broad-line wings arise in a non-relativistic bipolar jet. Two variations of the jet interpretation seem plausible: (1) a fast jet mixes 56Ni to high velocities in polar zones of the H-rich envelope; or (2) the reverse shock in the jet produces blue and red bumps in Balmer lines when a jet interacts with dense CSM. Jet-driven Type II SNe are predicted for collapsars resulting from a wide range of initial masses above 25 M⊙, especially at subsolar metallicity. This seems consistent with the SN host environment, which is either an extremely low-luminosity dwarf galaxy or the very remote parts of an interacting pair of star-forming galaxies. It also seems consistent with the apparently low 56Ni mass that may accompany black hole formation. We speculate that the jet survives to produce observable signatures because the star's H envelope was very low mass, having been mostly stripped away by the previous eruptive mass-loss indicated by the Type IIn features in the spectrum.