Can there exist a non-constant continuous function that has a derivative of zero everywhere? Somebody told me that there exists a continuous function with a derivative of zero everywhere that is not constant.
I cannot imagine how that is possible and I am starting to doubt whether it's actually true. If it is true, could you show me an example? If it is not true, how would you go about disproving it?

In short, no.  Your friend may be misremembering examples of continuous nowhere-differentiable functions, e.g., the Weierstrass function or the boundary of the Koch snowflake.
I was gong to cite the mean value theorem, but Jason DeVito beat me to the punch.  To add to his answer, you might also consider the Picard-Lindelof theorem.  What would it mean for $f$ to adopt two different values, since the theorem asserts that its behavior is locally that of a constant function?
