High-spatial and energy resolution electron energy-loss spectroscopy (EELS) can be used for detailed characterization of localized and propagating surface-plasmon excitations in metal nanostructures, giving insight into fundamental physical phenomena and various plasmonic effects. Here, applying EELS to ultra-sharp convex grooves in gold, we directly probe extremely confined gap surface-plasmon (GSP) modes excited by swift electrons in nanometre-wide gaps. We reveal the resonance behaviour associated with the excitation of the antisymmetric GSP mode for extremely small gap widths, down to B5 nm. We argue that excitation of this mode, featuring very strong absorption, has a crucial role in experimental realizations of non-resonant light absorption by ultra-sharp convex grooves with fabricationinduced asymmetry. The occurrence of the antisymmetric GSP mode along with the fundamental GSP mode exploited in plasmonic waveguides with extreme light confinement is a very important factor that should be taken into account in the design of nanoplasmonic circuits and devices.