While it is well-established that giant-planet occurrence rises rapidly with host star metallicity, it is not yet clear if small-planet occurrence around late-type dwarf stars depends on host star metallicity. Using the Kepler Data Release 25 planet candidate list and its completeness data products, we explore planet occurrence as a function of metallicity in the Kepler field's late-type dwarf stellar population. We find that planet occurrence increases with metallicity for all planet radii Rp down to at least Rp~2R{Earth}, and that in the range 2R{Earth}<~Rp<~5R{Earth}, planet occurrence scales linearly with metallicity Z. Extrapolating our results, we predict that short-period planets with Rp<~2R{Earth} should be rare around early-M dwarf stars with [M/H]<~-0.5 or late-M dwarf stars with [M/H]<~+0.0. This dependence of planet occurrence on metallicity observed in the Kepler field emphasizes the need to control for metallicity in estimates of planet occurrence for late-type dwarf stars like those targeted by Kepler's K2 extension and the Transiting Exoplanet Survey Satellite. We confirm the theoretical expectation that the small-planet occurrence-host star metallicity relation is stronger for low-mass stars than for solar-type stars. We establish that the expected solid mass in planets around late-type dwarfs in the Kepler field is comparable to the total amount of planet-making solids in their protoplanetary disks. We argue that this high efficiency of planet formation favors planetesimal accretion over pebble accretion as the origin of the small planets observed by Kepler around late-type dwarf stars.