Suppose there exists a subset of $\Bbb R$ which has cardinality $\omega_1$. Is it then necessarilly true that for every collection of $\omega_1$ subsets of $\Bbb R$ there exists a choice function?

I suppose that the answer for the above question is no, just like existence of countable subsets of $\Bbb R$ doesn't imply countable choice for $\Bbb R$. However, I suppose, every proof would either need explicit use of forcing, or a model of ZF which already has some weird properties, like amorphous sets or something.

Note that, obviously, we assume that AC doesn't necessarilly hold.