If $S$ is an uncountable subset of $C[0,1]$, then there is a uniformly convergent sequence $\{f_n\}$ of distinct functions of $S$.
I know how to do this for $C^1[0,1]$ since $S \subset \cup_{m,n \ge 0} \{f: \sup_{[0,1]} |f(x)| \le m, \sup_{[0,1]} |f'(x)| \le n \}$ and then $\{f: \sup_{[0,1]} |f(x)| \le m, \sup_{[0,1]} |f'(x)| \le n \}$ must be uncountable for some $m,n$ (since otherwise $S$ would be countable) and so contains a uniformly convergent sequence by Arzelà–Ascoli.