  We perform a search for transiting planets in the NASA K2 observations of the
globular cluster (GC) M4. This search is sensitive to larger orbital periods
($P\lesssim 35$ days, compared to the previous best of $P\lesssim 16$ days)
and, at the shortest periods, smaller planet radii (R$_p\gtrsim0.3$ R$_J$,
compared to the previous best of R$_p\gtrsim0.8$ R$_J$) than any previous
search for GC planets. Seven planet candidates are presented. An analysis of
the systematic noise in our data shows that most, if not all, of these
candidates are likely false alarms. We calculate planet occurrence rates
assuming our highest significance candidate is a planet and occurrence rate
upper limits assuming no detections. We calculate 3$\sigma$ occurrence rate
upper limits of 6.1\% for 0.71-2 R$_J$ planets with 1-36 day periods and 16\%
for 0.36-0.71 R$_J$ planets with 1-10 day periods. The occurrence rates from
Kepler, TESS, and RV studies of field stars are consistent with both a
non-detection of a planet and detection of a single hot Jupiter in our data.
Comparing to previous studies of GCs, we are unable to place a more stringent
constraint than Gilliland et al. 2000 for the radius-period range they were
sensitive to, but do place tighter constraints than both Weldrake et al. 2008
and Nascimbeni et al. 2012 for the large-radius regimes to which they were
sensitive.
