A function space $C_{p}(X)$ without a condensation onto a $\sigma$-compact space

Assuming that the minimal cardinality of a dominating family in $\omega ^{\omega }$ is equal to $2^{\omega }$, we construct a subset $X$ of a real line $\mathbb{R}$ such that the space $C_{p}(X)$ of continuous real-valued functions on $X$ does not admit any continuous bijection onto a $\sigma$-compact space. This gives a consistent answer to a question of Arhangel'skii.