Promise not to fall: Not to x Don't

The definition of DON´T is a command or entreaty not to do something (Webster´s Dictionary)
https://www.merriam-webster.com/dictionary/don't

As you are seeking guarantees here, not an order it´s signaled by "not to do", it´s less overbearing/demanding. It let´s open to the possibility of "not doing" (promise not fall - you may fall or not).
It´s a bit similar to "or not?" that leaves open to another possibility that is yes/yes sir!

https://dictionary.cambridge.org/gramma ... tences/not

https://www.easypacelearning.com/all-le ... ish-lesson

In this case, you can´t positively promise that you won´t fall/are not falling (anytime). Indeed, the person is asking you to "try not to fall". Again, there is another possibility open here, that of falling...