Example of a closed subspace of a Banach space which is not complemented?

Try the following article "A Survey of the Complemented Subspace Problem": https://arxiv.org/abs/math/0501048v1

Your suspicion about $c_0$ is correct. A couple of other examples: The disc algebra (those functions in $C(\mathbb{T})$ which are restrictions of functions analytic in the open unit disc) is closed in $C(\mathbb{T})$ but not complemented. Similarly, in $L^1(\mathbb{T})$, the subspace $H^1(\mathbb{T})$ consisting of functions whose negative Fourier coefficients vanish is closed but not complemented. See Rudin's Functional Analysis (the proof isn't very easy).

The article

Robert Whitley, Projecting $m$ onto $c_0$, The American Mathematical Monthly, Vol. 73, No. 3 (Mar., 1966), pp. 285-286

provides a short proof that $c_0$ is not complemented in $\ell^{\infty}$ by showing that $\ell^{\infty}/c_0$ does not have a countable set $f_n$ of continuous linear functions isolating zero (i.e. $\cap_n\ker f_n=\{0\}$).