= Solution
For integers $m,n\ge1$, let $E_{m,n}$ contain those $f\in C([0,1])$ for which there is some $x\in[0,1]$ such that
$$
|f(y)-f(x)|\le m|y-x|
$$
whenever $|y-x|<1/n$. Compactness of $[0,1]$ shows that $E_{m,n}$ is closed in the <uniform norm>: for a uniformly convergent sequence, choose convergent subsequences of the corresponding witness points and pass to the limit.
Each $E_{m,n}$ has empty interior. Given $f$ and a uniform-error tolerance, first approximate $f$ by a polygonal function, then add a sufficiently small-amplitude, sufficiently high-frequency sawtooth whose slopes dominate both $m$ and every slope of the polygonal approximation. At every point, one can move a distance below $1/n$ along one adjacent linear segment and obtain a difference quotient of magnitude greater than $m$. The perturbed function lies in the prescribed ball but outside $E_{m,n}$.
If $f$ is differentiable at some point, its nearby difference quotients are bounded, so $f\in E_{m,n}$ for some $m,n$. The functions differentiable somewhere therefore form a <meagre set>. By the <Baire category theorem>, their complement is nonempty; indeed it is dense. Thus
$$
\boxed{C([0,1])\text{ contains a nowhere differentiable continuous function}}.
$$
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