Smoothly slice links in $\mathbb{CP}^2 \# \overline{\mathbb{CP}^2}$

We show that there exists a link with 2 components which is not smoothly slice in $\mathbb{CP}^2 \# \overline{\mathbb{CP}^2}$. By contrast, it is well-known that every knot (i.e., link with 1 component) is smoothly slice therein. Our proof uses classical topological and smooth obstructions, as well as constructive arguments to exploit the symmetries of the problem. As a consequence, we show that there are infinitely many integer homology 3-spheres such that if any of them bounds a ribbon integer homology 4-ball, than there exists an exotic $\mathbb{CP}^2 \# \overline{\mathbb{CP}^2}$.