I felt the answer was no from the time I read your question, and now I think I have an argument to give you.  This comes from the context of studying $BV$ functions, where the notion of approximate gradient is important.  

http://www.encyclopediaofmath.org/index.php/Approximate_differentiability

The approximate gradient is the measure theoretic equivalent of the differential, defined as the object such that the difference quotients are small on a set of full or large measure.  In particular, every Sobolev function is approximately differentiable and the weak derivative is a candidate for the approximate differential, unique up to a set of Lebesgue measure zero.  But the strong derivative is also a candidate, and so they coincide, up to a set of Lebesgue measure zero.

We therefore overcome the difficulty of not assuming $f \in C^1(U)$ or Lipschitz by using a definition which is not for integrable functions (approximate differentiability).