= Solution
The theory $\operatorname{ACF}_0$ of <algebraically closed field>[algebraically closed fields] of characteristic zero is strongly minimal in its field sort. Let
$$
M=\overline{\mathbb Q(t_0,t_1,\ldots)}
$$
and take $\kappa=\aleph_1$. The countable tuples
$$
\bar a=(t_1,t_2,\ldots),\qquad
\bar b=(t_0,t_1,\ldots)
$$
have the same type because both are algebraically independent sequences. The element $t_0$ is independent from $\bar a$, but there is no element of $M$ independent from $\bar b$, since $\bar b$ is a transcendence basis and $M=\operatorname{acl}(\bar b)$. Thus the partial elementary map $\bar a\mapsto\bar b$ cannot be extended to $t_0$, so $M$ is not $\aleph_1$-homogeneous.
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