Smoothly slice links in S² × S² or $CP^2 # CP^2̄$

For X = S² × S² and CP² # CP²̄, we show that there exists a link with 2 components which is not smoothly slice in X. By contrast, it is well-known that every knot (i.e., link with 1 component) is smoothly slice in both S² × S² and CP² # CP²̄. Our proof uses classical topological and smooth obstructions, as well as constructive arguments to exploit the symmetries of the problem. As a side note, we also show that for every compact 4-manifold there exists a link that is not slice in it (either smoothly or topologically).