Does every almost-disjoint family of countably infinite sets whose pairwise intersections all have size ≠ 1 have Property B?

Trivial case: pairwise disjoint families.

If the A_i are pairwise disjoint (all intersections are empty, which in particular satisfies |A_i ∩ A_j| ≠ 1), then Property B holds trivially.

Proof sketch: Since each A_i is infinite, it has (at least) two distinct elements a_i and b_i. We can define a colouring that assigns colour 0 to a_i and colour 1 to b_i for each i (using disjointness, these choices don't conflict), and extend arbitrarily elsewhere. Then no A_i is monochromatic.