"No, I am not waking up."
So the thing is that "waking up" is more about getting up and out of bed in Japanese, not just becoming conscious after sleep. My wife (Japanese) gives me such a hard time for "not waking up" while I'm literally talking with her and she knows I've been conscious for a while, just reading in bed before "getting up". Think of it more as "getting up" than "waking up", although Japanese doesn't makr that distinction.
That's not an ending for the verb 起きる/おきる. じゃない is itself a verb (of sorts, an expression that functions similar to a verb) meaning "to not be". It's an opposite of です (to be). So, your construction might come across like "I is not wake up" instead of the intended "I won't wake up"
I know which question you mean. I think "No, I am not up" was a suggested translation by someone in the comments and in my opinion, it is incorrect and should not have been accepted.
In Japanese, "I am not up" is translated to 「(私は)起きていません」which means "I (私) do not exist (いません) as having woken up (起きて)".
「おきません」 is simple present tense, and Japanese simple present tense has three different usages: for general actions, for habitual actions, and for near future actions.
So the possible translations of 「(私は)起きません」 are "I generally don't wake up", "I don't have a habit of waking up", or "I am not going to wake up (soon)". Without any other context, the third option is probably the most common and so it should be the assumed meaning here.
So, I went back to the other question to check, and it does indeed accept "No, I am not up" (as of 30 Jul 17) as a correct translation of 「いいえ、おきません」.
I've reported it, as I believe it is blatantly incorrect. Frankly, I'm pretty disappointed in the course creators for letting that one through. I mean, a poorly structured and confusing course, and not anticipating all possible correct translations, I can forgive, but this is a pretty straightforward failure to understand the usage of simple present and continuous present tenses.
No, Japanese verbs don't work that way. ではありません is a full verb all on its own; it's the negative form of です, also a full verb on its own.
On the other hand, おきます and おきません are complete verbs, but おき is only the verb stem which cannot be completed by です and its various forms since you essentially end up with a word that is one and a half verbs long.