Locally trivial torsors that are not Weil-Ch\^atelet divisible

For every prime p we give infinitely many examples of torsors under abelian varieties over Q that are locally trivial but not divisible by p in the Weil-Ch^atelet group. We also give an example of a locally trivial torsor under an elliptic curve over Q which is not divisible by 4 in the Weil-Ch^atelet group. This gives a negative answer to a question of Cassels.