There are 'canonical' examples of maps on operator spaces which are not completely bounded. Nevertheless, I couldn't produce any examples of bounded projections on relatively easy to understand operator spaces which are not c.b. In particular, I failed this task for $\mathcal{K}(H)$ and $\mathcal{B}(H)$, where $H$ is a Hilbert space (I tried projections onto certain subspaces isomorphic to $c_0$ and $\ell_\infty$, respectively).

I would appreciate any examples.