Suppose that $X$ is a separable, infinite dimensional Banach space.
We say that a set of functions $\{f_\alpha\}_{\alpha \in A}$ separates points in $X$ if for every $x,y \in X$, there is an $\alpha$ such that $f_\alpha(x) \neq f_\alpha(y)$.
Is it the case that $C_c(X)$ separates points in $X$?
I have made some attempts to construct functions in $C_c(X)$ that separate distinct points $x,y$ in $X$ using the characterisation as compact subsets of $X$ as exactly the closed, bounded and flat subsets (where $K$ is flat if for every $\varepsilon > 0$ there is a finite dimensional subspace $F$ of $X$ such that $K \subseteq F + B(0,\varepsilon)$) but these attempts ultimately didn't get anywhere.
If it is helpful, I am really interested in the case where $X$ is a separable subspace of a Besov space $B_{\infty,\infty}^\alpha$ so would be happy with an answer using any other properties of that space.
The answer is negative since $C_c(X) = \{0\}$. Indeed, if $\phi \in C_c(X)$ satisfies $\phi(x) \ne 0$ for some $x \in X$, you have $$ \phi(y) \ne 0 \forall y \in X : \|y - x\| \le \varepsilon $$ for some small $\varepsilon > 0$. Hence, the support of $\phi$ contains a ball and, thus, is not compact.