  Extending M.\ Daws' definition of ultra-amenable Banach algebras, we
introduce the notion of operator ultra-amenability for completely contractive
Banach algebras. For a locally compact group $G$, we show that the operator
ultra-amenability of $A(G)$ imposes severe restrictions on $G$. In particular,
it forces $G$ to be a discrete, amenable group with no infinite abelian
subgroups. For various classes of such groups, this means that $G$ is finite.
