Building a 4-bit S-box with a 4-uniform DDT

I am looking for a 4-bit S-box which has only 4s in its difference distribution table (except the top left corner) but I was not able to find one in the literature.

Has such an S-box been already published? If not, is it theoretically possible to reach this property? If so, how to proceed?

Yes. All $$4\times4$$ bit S-boxes have been classified. It is possible to reach a maximum differential of 4 as well as optimum nonlinearity.

SERPENT used optimal $$4\times 4$$ S-boxes with respect to differential cryptanalysis. So does PRESENT, to the best of my knowledge.

in Blondeau-Nyberg which originally appeared in Eurocrypt 94 Chabaud-Vaudenay that such a DDT pattern would yield weak S-boxes against linear cryptanalysis. Here $$n=m=4,$$ $$cor_x(a\cdot x \oplus b\cdot F(x))$$ is the correlation bias of the corresponding linear characteristic, and $$P[\delta \stackrel{F}{\rightarrow} \Delta]$$ is the diferential probability.