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We give an example of a compact space X whose iterated continuous function spaces $C_p\left(X\right)$, $C_p{C}_p\left(X\right),...$ are Lindelöf, but X is not a Corson compactum. This solves a problem of Gul’ko (Problem 1052 in [11]). We also provide a theorem concerning the Lindelöf property in the function spaces $C_p\left(X\right)$ on compact scattered spaces with the ${\omega }_1$th derived set empty, improving some earlier results of Pol [12] in this direction.