Não, sua espada foi feita de aço valiriano.
White walkers can be killed using Valyrian steel as well. As per the show John himself was not aware of it till the time he was able to kill whitewalker with his sword.
Ele ganhou de Jeor, o Senhor Comandante da Patrulha da Noite antes dele.
Como por GOT Wiki:
Valyrian steel is recognizable from its sharpness, as well as a distinctive rippled pattern visible in blades made from it. Valyrian steel is one of the few things that can kill White Walkers, alongside dragonglass, although this property is not widely known, apparently, not even to the White Walkers themselves.