For a Hausdorff zero-dimensional topological space X and a totally ordered field F with interval topology, let $$C_c(X,F)$$ be the ring of all F-valued continuous functions on X with countable range. It is proved that if F is either an uncountable field or countable subfield of $${\mathbb {R}}$$ , then the structure space of $$C_c(X,F)$$ is $$\beta _0X$$ , the Banaschewski Compactification of X. The ideals $$\{O^{p,F}_c:p\in \beta _0X\}$$ in $$C_c(X,F)$$ are introduced as modified countable analogue of the ideals $$\{O^p:p\in \beta X\}$$ in C(X). It is realized that $$C_c(X,F)\cap C_K(X,F)=\bigcap _{p\in \beta _0X{\setminus } X} O^{p,F}_c$$ , and this may be called a countable analogue of the well-known formula $$C_K(X)=\bigcap _{p\in \beta X{\setminus } X}O^p$$ in C(X). Furthermore, it is shown that the hypothesis $$C_c(X,F)$$ is a Von-Neumann regular ring is equivalent to amongst others the condition that X is a P-space.