Let $X$ be an abelian scheme over some base scheme $S$. Even without projectivity hypothesis, we may talk of the dual abelian scheme $\hat X$ of $X$. Given a line bundle $\mathcal L$ over $X$, the associated Mumford line bundle $\Lambda(\mathcal L) := \mu^*\mathcal L \otimes \mathrm{pr}_1^*\mathcal L^{-1}\otimes \mathrm{pr}_2^*\mathcal L^{-1}$ on $X\times_S X$ induces a homomorphism $\lambda_{\mathcal L}:X\rightarrow \hat X$. We say that the line bundle $\mathcal L$ is non-degenerate when the homomorphism $\lambda_{\mathcal L}$ is an isogeny.
Is there always a non-degenerate line bundle $\mathcal L$ on $X$ ?
When the abelian scheme $X\rightarrow S$ is projective, any relatively ample line bundle on $X$ will do. In particular, this settles down the case where $S$ is noetherian normal, as any abelian scheme (even abelian algebraic space) over such an $S$ is automatically a projective scheme (see Faltings and Chai's book "Degeneration of Abelian Varieties", Remarks 1.10.a).
But what about the very general case, with no hypothesis on $S$ ?