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).

Suppose $f \in W^{1,1}_{loc}(U)$. Then no, since for such an $f$, we have that $Df$ exists and the approximate limit

$ap\lim_{y\to x} \frac{f(x)-f(y)-Df(x)(x-y)}{|x-y|} = 0$

exists for almost every $x$, while from assuming classical differentiability we have

$\lim_{y\to x} \frac{f(x)-f(y)-\nabla f(x)(x-y)}{|x-y|} = 0$

exists for every $x \in U$. In particular, the classical differential is a candidate for the approximate differential, and so $Df=\nabla f$ wherever the two exist, and hence in $U$ up to a set of measure zero.

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