The Li-rich stars are a class of rare objects with a surface lithium abundance, A(Li), that exceeds that of other stars in the same evolutionary stage. The origin of these stars is still debated, and valuable routes to look at include the Cameron-Fowler mechanism, a mass-transfer process in a binary system, or the engulfment of rocky planets or brown dwarfs. Metal-poor ([Fe/H]<−1 dex) stars are only a small fraction of the entire population of Li-rich stars. We observed the metal-poor ([Fe/H]=−3.95±0.11 dex) giant star HE 0057–5959 with MIKE at the Magellan Telescope, deriving A(Li)NLTE=+2.09±0.07 dex. Such an Li abundance is significantly higher, by about 1 dex, than that of other stars at the same evolutionary stage. A previous analysis of the same target suggested that its high A(Li) reflects an ongoing first-dredge-up process. We revised the nature of HE 0057-5959 by comparing its stellar parameters and A(Li) with appropriate stellar evolution models describing Li depletion due to the deepening of the convective envelope. This comparison rules out that HE 0057-5959 is caught during its first dredge-up, the latter having already ended according to the parameters of this star. Its A(Li), remarkably higher than the typical lithium plateau drawn by similar giant stars, demonstrates that HE 0057-5959 joins the class of the rare metal-poor, Li-rich stars. HE 0057-5959 is the most metal-poor, Li-rich star discovered so far. We considered different scenarios to explain this star also comparing it with the other metal-poor, Li-rich stars. No internal mixing able to activate the Cameron-Fowler mechanism is known for metal-poor stars at this evolutionary stage. The engulfment of planets is also disfavoured because such metal-poor stars should not host planets. Finally, HE 0057-5959 is one of the most Na-rich among the Li-rich stars, and we found that a strong excess of Na abundance is common to all three Li-rich stars with [Fe/H]<–3 dex. This finding could support the scenario of mass transfer from a massive companion star (able to simultaneously produce large amounts of both elements) in a binary system, even if we found no evidence of radial velocity variations.