What if Cp(X) is perfectly normal?

We prove that Cp(X) × Cp(X) is perfectly normal iff for any natural n ⩾ 1 all closed subspaces of Xn are separable. It is established that the existence of a perfectly normal non-Lindelöf Cp(X) is independent of ZFC. We show that it is consistent with ZFC that Cp(X) is perfectly normal iff (Cp(X))ω is hereditarily Lindelöf. The situation when all closed subsets of Cp(X) are separable is also dealt with.