Does there exist a differentiable conservative vector field with non - vanishing curl?

I think you are referring to counterexamples in the context of Clairaut's theorem, i.e., examples of functions where the order of partial derivatives may not be interchanged. Wikipedia has an example in two variables with a nice picture.

One could adapt such an example to establish nonvanishing curl at a single point: add a dummy third variable $z$ and consider the gradient of that function. A gradient is always conservative, but the $z$-component of the curl of that gradient is nonzero at the origin.

Such examples do not work for constructing a nonvanishing curl in the entire domain, however; distribution theory shows that partial derivatives can be freely interchanged up to a set of volume 0.