Aims: We describe the discovery of an extremely wide pair of low-mass stars with a common large proper motion and discuss their possible membership in a Galactic halo stream crossing the Solar neighbourhood. Methods: In a high proper motion survey of the southern sky we used multi-epoch positions and photometry from the SuperCOSMOS Sky Surveys. New nearby ultracool dwarf and subdwarf candidates were selected among the faint and red high proper motion objects, and subsequently confirmed by low-resolution classification spectroscopy. The resulting spectroscopic distance estimates, approximate radial velocity measurements and improved proper motions involving additional epochs from the Two Micron All Sky Survey and from the DEep Near-Infrared Survey were used to compute Galactic space velocities. Results: The late-type (M 7) dwarf SSSPM J2003-4433 and the ultracool subdwarf SSSPM J1930-4311 (sdM 7) sharing the same very large proper motion of about 860 mas/yr were found in the same sky region with an angular separation of about 6°. From the comparison with other high proper motion catalogues we have estimated the probability of a chance alignment of the two new large proper motions to be less than 0.3%. From the individually estimated spectroscopic distances of about 38+10-7 pc and 72+21-16 pc, respectively for the M 7 dwarf and the sdM 7 subdwarf, and in view of the accurate agreement in their large proper motions we assume a common distance of about 50 pc and a projected physical separation of about 5 pc. The mean heliocentric space velocity of the pair (U,V,W)=(-232, -170, +74) km s-1, based on the correctness of the preliminary radial velocity measurement for only one of the components and on the assumption of a common distance and velocity vector, is typical of the Galactic halo population. Conclusions: The large separation and the different metallicities of dwarfs and subdwarfs make a common formation scenario as a wide binary (later disrupted) improbable, although there remains some uncertainty in the spectroscopic classification scheme of ultracool dwarfs/subdwarfs so that a dissolved binary origin cannot be fully ruled out yet. It seems more likely that this wide pair is part of an old halo stream. Higher-resolution spectroscopic observations are needed to measure accurate radial velocities of both components. Further, we suggest to check the M 7 dwarf for an unresolved binary status, which would explain its shorter spectroscopic distance estimate, and to place both objects on a trigonometric parallax program. Based on observations with the ESO 3.6 m/EFOSC2 at the European Southern Observatory, La Silla (ESO program 70.C-0568).