For ${\cal H}$ an infinite-dimensional Hilbert space, the canonical projection $\cal{B(H)}\rightarrow \cal{A(H)}$ (where $\cal{A(H)}$ is the Calkin algebra), has no continuous, linear section; this is due to E.O. Thorp, [Link](https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-10/issue-2/Projections-onto-the-subspace-of-compact-operators/pjm/1103038424.full)