This is (unfortunately for you) well-known. Theorem 10.1 of this paper shows that over an algebraically closed field of characteristic zero the Mordell-Weil rank of any positive-dimensional abelian variety is infinite, and (as remark 1 below the theorem mentions) this implies that for any abelian variety over field of characteristic zero, there are finite extensions over which the ranks get arbitrarily large.

The result also holds for fields of positive characteristic which are not algebraic over a finite field.

This is (unfortunately for you) well-known. Theorem 10.1 of Frey, Gerhard; Jarden, Moshe, Approximation theory and the rank of Abelian varieties over large algebraic fields, Proc. Lond. Math. Soc., III. Ser. 28, 112-128 (1974). ZBL0275.14021 shows that over an algebraically closed field of characteristic zero, the Mordell-Weil rank of any positive-dimensional abelian variety is infinite. And as Remark 1 below the theorem mentions, this implies that for any abelian variety over field of characteristic zero, there are finite extensions over which the ranks get arbitrarily large.

The result also holds for fields of positive characteristic which are not algebraic over a finite field. (A non-firewalled copy of the paper is available here.)