Tsunamis are triggered by sudden seafloor displacements, and usually originate from seismic activity at faults. Nevertheless, strike-slip faults are usually disregarded as major triggers, as they are thought to be capable of generating only moderate seafloor deformation; accordingly, the tsunamigenic potential of the vertical throw at the tips of strike-slip faults is not thought to be significant. We found the active dextral NW-SE Averroes Fault in the central Alboran Sea (westernmost Mediterranean) has a historical vertical throw of up to 5.4 m at its northwestern tip corresponding to an earthquake of Mw 7.0. We modelled the tsunamigenic potential of this seafloor deformation by Tsunami-HySEA software using the Coulomb 3.3 code. Waves propagating on two main branches reach highly populated sectors of the Iberian coast with maximum arrival heights of 6 m within 21 and 35 min, which is too quick for current early-warning systems to operate successfully. These findings suggest that the tsunamigenic potential of strike-slip faults is more important than previously thought, and justify the re-evaluation of tsunami early-warning systems worldwide.