Let . The product space is separable, as is the product space . By a theorem of [Ross & Stone], the product of continuum many separable spaces is a separable space. However, both and are not hereditarily separable. In a previous post on the first uncountable ordinal, I showed that the space of continuous functions on is not separable (see the post). I show in this post that the product space contains a copy of this function space. Hence both and are not hereditarily separable.
In my previous post, I showed that is not separable. Note that contains as a subspace. Thus is not hereditarily separable. Note that is considered a subspace of . Thus is not hereditarily separable. Note that is considered a subspace of . If the cardinality of is greater than , then all functions in that are zero on all points of would be a copy of . If the cardinality of equals , Thus both and are not hereditarily separable.
[Ross & Stone]
Ross, K. A. and Stone, A. H.  Products of Separable Spaces, The American Mathematical Monthly, Vol 71, No. 4, pp. 398-403.