Context. Gamma-ray binaries are systems composed of a massive star and a compact object that exhibit emission from radio to very high energy gamma rays. They are ideal laboratories to study particle acceleration and a variety of physical processes that vary as a function of the orbital phase. Aims. We aim to study the radio emission of the gamma-ray binary 1FGL J1018.6–5856 to constrain the emitting region and determine the peculiar motion of the system within the Galaxy to clarify its origin. Methods. We analyzed an observation of 1FGL J1018.6–5856 with the Australian Long Baseline Array (LBA) at 8.4 GHz to obtain an accurate astrometry of the system and study its emission on milliarcsecond scales. We combined these data with the optical Gaia DR2 and UCAC4 catalogs to consolidate the astrometry information therein. Results. The gamma-ray binary 1FGL J1018.6–5856 shows compact radio emission (< 3 mas or ≲20 au at ∼6.4 kpc distance), implying a brightness temperature of ≳5.6 × 10⁶ K, and confirming its nonthermal origin. We report consistent results between the proper motion reported by Gaia DR2 and the positions obtained from the Gaia DR2, UCAC4, and LBA data (spanning 20 yr in total). We also determined the distance to 1FGL J1018.6–5856 to be 6.4_−0.7^+1.7. Together with the radial velocity of the source we computed its three-dimensional (3D) proper and peculiar motion within the Galaxy. We obtained a peculiar motion of 1FGL J1018.6–5856 on its regional standard of rest (RSR) frame of |u| = 45₋₉⁺³⁰, with the system moving away from the Galactic plane. In the simplest scenario of a symmetric stellar core collapse we estimate a mass loss of 4 ≲ ΔM ≲ 9 M_⊙ during the creation of the compact object. Conclusions. 1FGL J1018.6–5856 exhibits compact radio emission similar to that detected in other gamma-ray binaries. We provide the first accurate peculiar motion estimations of the system and place it within the Galaxy. The obtained motion and distance excludes the physical relation of the binary source with the supernova remnant (SNR) G284.3−1.8.