Aims. We study the effect a guiding magnetic field has on the formation and structure of a pair jet that propagates through a collisionless electron–proton plasma at rest.Methods. We model with a particle-in-cell (PIC) simulation a pair cloud with a temperature of 400 keV and a mean speed of 0.9c (c - light speed). Pair particles are continuously injected at the boundary. The cloud propagates through a spatially uniform, magnetized, and cool ambient electron–proton plasma at rest. The mean velocity vector of the pair cloud is aligned with the uniform background magnetic field. The pair cloud has a lateral extent of a few ion skin depths.Results. A jet forms in time. Its outer cocoon consists of jet-accelerated ambient plasma and is separated from the inner cocoon by an electromagnetic piston with a thickness that is comparable to the local thermal gyroradius of jet particles. The inner cocoon consists of pair plasma, which lost its directed flow energy while it swept out the background magnetic field and compressed it into the electromagnetic piston. A beam of electrons and positrons moves along the jet spine at its initial speed. Its electrons are slowed down and some positrons are accelerated as they cross the head of the jet. The latter escape upstream along the magnetic field, which yields an excess of megaelectronvolt positrons ahead of the jet. A filamentation instability between positrons and protons accelerates some of the protons, which were located behind the electromagnetic piston at the time it formed, to megaelectronvolt energies.Conclusions. A microscopic pair jet in collisionless plasma has a structure that is similar to that predicted by a hydrodynamic model of relativistic astrophysical pair jets. It is a source of megaelectronvolt positrons. An electromagnetic piston acts as the contact discontinuity between the inner and outer cocoons. It would form on subsecond timescales in a plasma with a density that is comparable to that of the interstellar medium in the rest frame of the latter. A supercritical fast magnetosonic shock will form between the pristine ambient plasma and the jet-accelerated plasma on a timescale that exceeds our simulation time by an order of magnitude.Key words: plasmas / methods: numerical / acceleration of particles / X-rays: binaries⋆ Movies associated to Figs. 4 and 8 are available at https://www.aanda.org