Listing examples is easy in Banach space theory.
1) (1950) R.C.James' construction of non-reflexive (Banach) space isometric to its bidual.
2) (1974) R.C.James' construction of a space whose dual is non-separable and all of its subspaces contain a copy of $\ell_2$.
3) (1993) Gowers and Maurey's construction of a space such that no infinite dimensional subspace has a non-trivial complemented subspace (hereditarily indecomposable).
4) (2010) Argyros and Haydon's construction of a space on which every operator is a scalar multiple of the identity plus a compact operator (see Gowers' blog).
I'm not convinces these all meet the criterion of being unexpected. In the case of (4), people expected a construction was possible, but for a long time nothing seemed to work and some very bright people had worked on the problem.