For a topological space $X$ let $C_c(X)$ be the space of compactly supported continuous real-valued functions. It is well known that $C_c(X)$ (provided with the sup norm) is separable, if $X$ is a locally compact metric space which is also $\sigma$-compact. See, for example Separability of functions with compact support.
I'm looking for examples of metric spaces $X$ such that $C_c(X)$ is not separable. While it's easy to find examples when $X$ is not $\sigma$-compact (see below), I haven't been able to find an example where $X$ is $\sigma$-compact, but not locally compcat. The space $\mathbb{R}^{(\infty)}$ of all finite real-valued sequences (sup norm) is $\sigma$-compact but not loccaly compact. But I haven't been able to check if $C_c(\mathbb{R}^{(\infty)})$ is separable or not.
Question 1: Is there an example of a $\sigma$-compact metric space $X$ such that $C_c(X)$ is not separable?
Question 2: Is $C_c(\mathbb{R}^{(\infty)})$ separable?
Case $X$ not $\sigma$-compact: For the sake of completness let me give an example for the case when $X$ is locally compact but not $\sigma$-compact. Let $I$ be an uncountable index set of compact, locally compact metric spaces $(K_i,d_i)$ with diameter at most 1. Give the disjoint union $X=\coprod_{i\in I}K_i$ the metric $d(x,y)=d_i(x,y)$ if there is $i\in I$ such that $x,y\in K_i$ and $d(x,y)=2$ otherwise. $K_i$ is compact in $X$ and in particular, $X$ is locally compact. If $K$ is a compact subset of $X$, the set $I(K)=\{i\in I\mid K\cap K_i\neq \emptyset\}$ is finite (as one can see by covering $K$ with open balls of radius 1/2). Now, if $\{f_n\mid n\in\mathbb{N}\}$ is any countable subset of $C_c(X)$, let $K_n$ be the support of $f_n$. Hence, for $i\in I\setminus \cup_n I(K_n)$, each $f_n$ restricts to zero on $K_i$ and thus the characteristic function $1_{K_i}$ is in $C_c(X)$ and has distance at least 1 to each $f_n$. So no countable subset of $C_c(X)$ is dense. q.e.d.