  It is notoriously difficult to localize short $\gamma$-ray bursts (sGRBs) and
their hosts to measure their redshifts. These measurements, however, are
critical to constrain the nature of sGRB progenitors, their redshift
distribution and the $r$-process element enrichment history of the universe.
Here, we present spectroscopy of the host galaxy of GRB111117A and measure its
redshift to be $z = 2.211$. This makes GRB111117A the most distant
high-confidence short duration GRB detected to date. Our spectroscopic redshift
supersedes a lower, previously estimated photometric redshift value for this
burst. We use the spectroscopic redshift, as well as new imaging data to
constrain the nature of the host galaxy and the physical parameters of the GRB.
The rest-frame X-ray derived hydrogen column density, for example, is the
highest compared to a complete sample of sGRBs and seems to follow the
evolution with redshift as traced by the hosts of long GRBs (lGRBs). The host
lies in the brighter end of the expected sGRB host brightness distribution at
$z = 2.211$, and is actively forming stars. Using the host as a benchmark for
redshift determination, we find that between 43 and 71 per cent of all sGRB
redshifts should be missed due to host faintness for hosts at $z\sim2$. The
high redshift of GRB111117A is evidence against a lognormal delay-time model
for sGRBs through the predicted redshift distribution of sGRBs, which is very
sensitive to high-$z$ sGRBs. From the age of the universe at the time of GRB
explosion, an initial neutron star (NS) separation of $a_0 < 3.2~R_\odot$ is
required in the case where the progenitor system is a circular pair of
inspiralling NSs. This constraint excludes some of the longest sGRB formation
channels for this burst.
