  We present a deep Chandra X-ray Observatory study of the peculiar binary
radio millisecond pulsar PSR J1740--5340 and candidate millisecond pulsars
(MSPs) in the globular cluster NGC 6397. The X-rays from PSR J1740--5340 appear
to be non-thermal and exhibit variability at the binary period. These
properties suggest the presence of a relativistic intrabinary shock formed due
to interaction of a relativistic rotation-powered pulsar wind and outflow from
the unusual "red-straggler/sub-subgiant" companion. We find the X-ray source
U18 to show similar X-ray and optical properties to those of PSR J1740--5340,
making it a strong MSP candidate. It exhibits variability on timescales from
hours to years, also consistent with an intrabinary shock origin of its X-ray
emission. The unprecedented depth of the X-ray data allows us to conduct a
complete census of MSPs in NGC 6397. Based on the properties of the present
sample of X-ray--detected MSPs in the Galaxy we find that NGC 6397 probably
hosts no more than 6 MSPs.
