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) łesssim 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 $^56$Ni 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 $^56$Ni 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.