  We study $p$-adic families of cohomological automorphic forms for
${\mathrm{GL}}(2)$ over imaginary quadratic fields and prove that families
interpolating a Zariski-dense set of classical cuspidal automorphic forms only
occur under very restrictive conditions. We show how to computationally
determine when this is not the case and establish concrete examples of families
interpolating only finitely many Bianchi modular forms.
