"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.
"I will not go, not even next week." implies that you never intend to go, now or in the future.
"I will not go, at least not next week." implies that you will not go next week but you may go in the future.
This is why if you wrote "at least" you were marked incorrect.