We analyzed a deep XMM-Newton observation of the radio-quiet {gamma}-ray PSR J2055+2539. The spectrum of the X-ray counterpart is nonthermal, with a photon index of {Gamma}=2.36+/-0.14 (1{sigma} confidence). We detected X-ray pulsations with a pulsed fraction of 25%+/-3% and a sinusoidal shape. Taking into account considerations on the {gamma}-ray efficiency of the pulsar and on its X-ray spectrum, we can infer a pulsar distance ranging from 450 to 750 pc. We found two different nebular features associated with PSR J2055+2539 and protruding from it. The angle between the two nebular main axes is ~162.8{deg}+/-0.7{deg}. The main, brighter feature is 12' long and <20" thick, characterized by an asymmetry with respect to the main axis that evolves with the distance from the pulsar, possibly forming a helical pattern. The secondary feature is 250"x30". Both nebulae present an almost flat brightness profile with a sudden decrease at the end. The nebulae can be fitted by either a power-law model or a thermal bremsstrahlung model. A plausible interpretation of the brighter nebula is in terms of a collimated ballistic jet. The secondary nebula is most likely a classical synchrotron-emitting tail.