Is it possible to prove stronger bounds for the deterministic communication complexity compared to nondeterministic communication complexity?
It seems to me that there is some incoherency in the question. What do you mean by better?
Are you looking for a function where $N(f) \ll D(f)$ or a function where $D(f) \ll N(f)$? 

Deterministic protocols are a special case of nondeterministic protocols, so we always have $N(f) \leq D(f)$.

For examples of functions where $N(f) \ll D(f)$ have a look at the Nisan and Kushilevitz's book.