"Yo no voy, ni siquiera la próxima semana."
That's true, sorry, I wrote 'at least next week' didn't think about the not. 'At least not next week' would make sense. I suppose it depends on context. 'At least not next week' indicates there is a possible time the speaker would go, whereas 'not even next week' rules it out going entirely. From a literal translation point of view, I guess one would think either would be correct.