Let $X$, $Y$ be irreducible Noetherian schemes. Let $f:X\rightarrow Y$ be a proper morphism. Assume that for any $y\in Y$, the base change $X\times \mathrm{Spec}\,k(y)\rightarrow \mathrm{Spec}\, k(y)$ is a smooth projective morphism. Is $f$ projective?
EDIT (in response to abx): assume moreover that $Y$ is reduced and geometrically unibranch.