The Suslin line is a topological space which has the CCC property but is not separable.
However, proving that the Suslin line exists cannot be done within $ZFC$. Why?
Assuming $V=L$ (The axiom of constructibility) implies certain combinatorial properties from which we can construct a Suslin line, however assuming a different axiom $MA$ (Martin's axiom) we can prove that no Suslin line exists.
The result of this is that we cannot prove from ZFC alone that every CCC space is separable.
Added: (To make this answer complete, I'll add the right answer given by Henno Brandsma in the comments)
We cannot prove in ZFC that CCC spaces are separable because $\{0,1\}^X$ has CCC for any $X$, but is only separable for $|X|\le\frak c$. In particular, taking $X=P(\mathbb R)$ gives us a CCC space which is not separable.