An extremely wide and very low-mass pair with common proper motion. Is it representative of a nearby halo stream?

(abridged) 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. (...) The late-type (M7) dwarf SSSPM J2003$-$4433 and the ultracool subdwarf SSSPM J1930$-$4311 (sdM7) 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\degr. 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 M7 dwarf and the sdM7 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, 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. 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. (...)