If every partitioning of $X$ has a choice function, is $X$ necessarily well-orderable?

Question

Working over the ZF axioms, it's clear that if $X$ is a well-orderable set, then every partitioning of $X$ has a choice function, by choosing the minimum of each cell.

Question. Does the converse hold?

i.e. If every partitioning of $X$ has a choice function, is $X$ necessarily well-orderable?

Answer 1

No.

Suppose that $A$ is a strongly amorphous set. Namely,

1. $A$ is infinite Dedekind-finite set, such that every subset of $A$ is finite or co-finite.
2. Every partition has only finitely many non-singletons.

Now every function with domain $A$ induces a partition and only finitely many fibers have nontrivial choices.

Of course, it is consistent that there are strongly amorphous sets.