Existence of separable Fréchet–Urysohn C_p(X) not hereditarily F-separable

Construct a Tychonoff space X such that C_p(X), the space of all real-valued continuous functions on X with the topology of pointwise convergence, is separable and Fréchet–Urysohn, but C_p(X) is not hereditarily F-separable.

Background

The paper notes that, in some models of set theory, separable Fréchet–Urysohn C_p(X) coincide with first countable C_p(X), suggesting that existence of certain counterexamples may be model-dependent.

This question asks for a concrete example separating Fréchet–Urysohn (and separability) from hereditary F-separability in C_p(X), highlighting subtle interactions among these properties.