IIUC Remark 5.3.5 in EGA II says that there exist proper non-projective morphisms $X\rightarrow Y$ where $Y$ is the spectrum of a finite-dimensional $\mathbb{C}$-algebra such that the induced morphism $X_{red}\rightarrow Y_{red}$ is projective. Here is .pdf file.

May somebody give an example?

FWIW here some examples of finite-dimensional commutative associative unital $\mathbb{C}$-algebras are listed.