Context : With the discovery of a planetary system around the ultracool dwarf TRAPPIST-1 , there has been a surge of interest in such stars as potential planet hosts .
Planetary systems around ultracool dwarfs represent our best chance of characterising temperate rocky-planet atmospheres with JWST .
However , TRAPPIST-1 remains the only known system of its kind , and the occurrence rate of planets around ultracool dwarfs is still poorly constrained .

Aims : We seek to perform a complete transit search on the ultracool dwarfs observed by NASA ’ s K2 mission , and use the results to constrain the occurrence rate of planets around these stars .

Methods : We filter and characterise the sample of ultracool dwarfs observed by K2 , by fitting their spectral energy distributions , and using parallaxes from Gaia .
We build an automatic pipeline to perform photometry , detrend the lightcurves , and search for transit signals .
Using extensive injection-recovery tests of our pipeline , we compute the detection sensitivity of our search , and thus the completeness of our sample .
We infer the planetary occurrence rates within a hierarchical Bayesian model ( HBM ) to treat uncertain planetary parameters .
With the occurrence rate parametrised by a step-wise function , we show a convenient way to directly marginalise over the second level of our HBM ( the planetary parameters ) .
Our method is applicable generally and can greatly speed up inference with larger catalogues of detected planets .

Results : We detect one planet in our sample of 702 ultracool dwarfs : a previously-validated mini-Neptune .
We thus infer a mini-Neptune ( 2 - 4 \si { \Re } ) occurrence rate of \eta = 0.20 ^ { +0.16 } _ { -0.11 } within orbital periods of 1 - 20 days .
For super-Earths ( 1 - 2 \si { \Re } ) and ice/gas giants ( 4 - 6 \si { \Re } ) within 1 - 20 days , we place 95 % credible intervals of \eta < 1.14 and \eta < 0.29 , respectively .
If TRAPPIST-1-like systems were ubiquitous , we would have had a \sim 96 \% chance of finding at least one .

Conclusions :