For each global field $K$, can we always find an Abelian variety $A$ with $A(K)$ rank $0$?

For each global field $K$, can we always find an Abelian variety $A$ with $A(K)$ rank $0$?

By Lang-Neron, Mordell Weil is also true for finite type fields (fields finitely generated over their prime field). Can we also find Abelian varieties over these fields with rank 0?