That $C[0,1]$ fails the weak Banach Saks property appears as problem 17 in chapter VII in Joseph Diestel's Sequences and Series in Banach Spaces.
The problem gives an outline:
1) For $k$ a fixed positive integer, construct a nonnegative sequence $(g_n^k)_n$ in $B(C[0,1])$ satisfying:
- $g_n^k(t)=0$ if $t\notin ((k-1)/k, k/(k+1))$.
- $(g_n^k)$ converges pointwise to 0 on $[0,1]$.
- If $n_1, then there is an $a\in[0,1]$ with $g_{n_1}^k(a)=g_{n_2}^k(a)=\cdots=g_{n_k}^k(a)=1$.
2) Define $f_n=g_n^1+g_n^2+\cdots+g_n^n$. Show that:
- $(f_n)$ is weakly null in $C[0,1]$ (which is equivalent to saying $(f_n)$ converges pointwise to 0 and is bounded).
- If $n_1, then $(f_{n_1}+\cdots f_{n_{2m}})(t)\ge {1\over 2}$ for all $t\in[0,1]$ (I believe Diestel has a typo here, it should be $\ge m/2$ for some $t\in[0,1]$).
This was first proved by J. Schreier in Ein Gegenbeispiel zur Theorie der schwachen Konvergence, Studia Math. 2 (1930), 58–62. H. P. Rosenthal's article in volume 2 of Handbook of the Geometry of Banach Spaces (proposition 4.21, page 1585) may also be helpful.