**In keeping his self-made promise to go fast and not stop,** the hare took off and ran at top speed until he came to a broad river.

That is a perfectly well formed sentence in English.

**A promise to do something** is very idiomatic. "To" is used to show a purpose or goal.


Question"Is "promise going fast and not stopping" grammatically correct?

No, it is incorrect. The hare promised  himself to go fast and not stop.