'not one bit' is correct and is a phrase often used in colloquial English. It is typically used to emphasize the extent of something by adding the word 'not' before it. Example: "I am not one bit surprised that he already finished the project."
"Not one bit of suspicion fell on Molly or me," Calvin observes.
"Not one bit," Kelly says firmly.
Not one bit, as it transpired.
Not one bit of tobacco had landed anywhere.
The thrashing she got made her repent not one bit.
And I didn't miss them, not one bit.
I wouldn't argue with you, not one bit.
Writing is not one bit better for me there.
Writers' egos, famously large, concerned him not one bit.
As a baseball fan first, it wasn't one bit frustrating.
I understand that what cake-throwers do isn't one bit more authentic that I was.
Ludwig does not simply clarify my doubts with English writing, it enlightens my writing with new possibilities
Simone Ivan Conte
Software Engineer at Adobe, UK