Is the space $C[0,1]$ locally compact?

Consider the intervals $E_n=[1/n,1/(n+1)]$, $n\in\mathbb N$. Define continuous functions $f_n$ such that $f_n=0$ outside $E_n$, and $f_n=1$ on the mid point of $E_n$.

Then $\|f_n\|_\infty=1$ for all $n\in\mathbb N$, and $\|f_n-f_m\|_\infty=1$ whenever $n\ne m$, so the sequence admits no convergent subsequence.

Try $f_n(x):=x^n$.

The deeper reason is that $C[0,1]$ is infinite dimensional, so by Riesz theorem its unit ball is not compact.