We observed the young pulsar J1617-5055 with the Chandra ACIS detector for 60 ks. In addition to the pulsar, the X-ray images show a faint pulsar wind nebula (PWN) seen up to ∼1' from the pulsar. Deconvolution and reconstruction of the image reveal a brighter compact PWN component of ∼1'' size. The total PWN luminosity in the 0.5-8 keV band, Lpwn= (3.2-3.7)x1033 erg/s for d = 6.5 kpc, is a fraction of 2x10–4 of the pulsar's spin-down power and a fraction of 0.2 of the pulsar's X-ray luminosity, which is a factor of 20 lower than one would expect from an average empirical relation, Lpwn ∼ 4Lpsr. The pulsar's spectrum can be described by an absorbed power law with nH~ 3.5x1022/cm2 and Γ ~ 1.1, harder than any other pulsar spectrum reliably measured in the soft X-ray band. This nonthermal emission is ~50% pulsed, with one peak per period. We have also investigated a possible connection between J1617 and the extended TeV source HESS J1616-508 whose center is located 10' west of the pulsar. We find no preferential extension of the X-ray PWN toward the TeV source. Therefore, the Chandra data do not provide conclusive evidence for PSR J1617-5055 and HESS J1616-508 association. We have also analyzed archival X-ray, radio, and IR data on the HESS J1616-508 region and found traces of diffuse emission coinciding with the central part of HESS J1616-508. We speculate that the TeV source may be multiple, with most of the emission coming from an unknown supernova remnant or a star forming region, while some fraction of the TeV emission still may be attributed to the PWN around PSR J1617-5055.