INTEGRAL/SPI has recently observed a strong and extended emission resulting from electron-positron annihilation located in the Galactic center region, consistent with the Galactic bulge geometry, without any counterpart at high gamma-ray energies, nor in the 1809 keV $^{26}$Al decay line. In order to explain the rate of positron injection in the Galactic bulge, estimated to more than 10$^{43}$ s$^{-1}$, the most commonly considered positron injection sources are type Ia supernovae. However, SN Ia rate estimations show that those sources fall short to explain the observed positron production rate, raising a challenging question about the nature of the Galactic positron source. In this context, a possible source of Galactic positrons could be supernova events of a new type, as the recently observed SN2003dh/GRB030329, an exploding Wolf-Rayet star (type Ic supernova) associated with a hypernova/gamma-ray burst; the question about the rate of this kind of events remains open, but could be problematically low. In this paper, we explore the possibility of positron production and escape by such an event in the framework of an asymmetric model, in which a huge amount of $^{56}$Ni is ejected in a cone with a very high velocity; the ejected material becomes quickly transparent to positrons, which spread out in the interstellar medium.