The number of birds killed by the H5N1 strain has climbed to four as the virus is confirmed to have crossed the border into South Australia.

Testing is still under way on a second bird after Australia's first case of the deadly H5 bird flu was confirmed on WA's south coast.

The H5N1 variant of the virus was confirmed in a sick brown skua found on a remote beach near Esperance in Western Australia, about 700km southeast of Perth.