Let $K$ be a field and $G_K$ be its absolute Galois group. Let $E_1,E_2$ be two elliptic curves over $K$. Assume that there exists an isogeny $f:E_1\rightarrow E_2$. Let $p$ be a prime number. Then $f$ induces an isomorphism of rational Tate module $V_p(E_1)\cong V_p(E_2)$ as representations of $G_K$: let $f^{\vee}:E_2\rightarrow E_1$ be the dual isogeny, then $f\circ f^{\vee}:E_1\rightarrow E_1$ is $[\mathrm{deg}(f)]$ and on the Tate module $T_p(E_1)$ $f\circ f^{\vee}$ is just multiplicated by $\mathrm{deg}(f)$. So after tensoring $\mathbb{Q}_p$, $f$ induces an isomorphism and the isomorphism is $G_K$-equivariant.
My question is whether we can always get an isomorphism of integral Tate module $T_p(E_1)\cong T_p(E_2)$ as $G_K$-modules over $\mathbb{Z}_p$? Notice that the isomorphism doesn't need to be induced by $f$.
There is a possible way to find a counterexample. Tate's isogeny theorem tells that the isogeny classes of elliptic curves over a finite field $\mathbb{F}_q$ is determined by the rational representations. If we could construct elliptic curves over a finite field for any integral representations, then we just need to find two non-isomorphic integral models for a rational representations then we get a counterexample for the original question.
If the counterexample exists for general fields, can it be true for some special fields?