For an elliptic curve $E$ over $\mathbb{Q}$, it is well-known that the torsion points on $E$ are integral points.

Then, is it possible that there exists an example whose all of non-torsion rational points (or all of points of a subgroup of $E_{free}(\mathbb{Q}$)) are integral points (of course, with respect to affine coordinate)?

I never think that there is such an example and maybe it is a stupid question.