  The search for small planets orbiting late M dwarfs holds the promise of
detecting Earth-size planets for which their atmospheres could be characterised
within the next decade. The recent discovery of TRAPPIST-1 entertains hope that
these systems are common around hosts located at the bottom of the main
sequence. In this Letter, we investigate the ability of the repurposed Kepler
mission (K2) to probe planetary systems similar to TRAPPIST-1. We perform a
consistent data analysis of 189 spectroscopically confirmed M5.5 to M9 late M
dwarfs from campaigns 1-6 to search for planet candidates and inject transit
signals with properties matching TRAPPIST-1b and c. We find no transiting
planet candidates across our K2 sample. Our injection tests show that K2 is
able to recover both TRAPPIST-1 planets for 10% of the sample only, mainly
because of the inefficient throughput at red wavelengths resulting in
Poisson-limited performance for these targets. Increasing injected planetary
radii to match GJ1214b's size yields a recovery rate of 70%. The strength of K2
is its ability to probe a large number of cool hosts across the different
campaigns, out of which the recovery rate of 10% may turn into bona-fide
detections of TRAPPIST-1 like systems within the next two years.
