  It is shown that integral operators of the fully nonlinear type
$K(x)(t)=\int_\Omega k(t,s,x(t),x(s))\,ds$ exhibit similar degeneracy phenomena
in a large class of spaces as superposition operators $F(x)(t)=f(t,x(t))$. In
particular, $K$ is Fr\'echet differentiable in $L_p$ only if it is affine with
respect to the "$x(t)$" argument. Similar degeneracy results hold if $K$
satisfies a local Lipschitz or compactness condition. Also vector functions,
infinite measure spaces, and a much richer class of function spaces than only
$L_p$ are considered. As a side result, degeneracy assertions for superposition
operators are obtained in this more general setting, complementing the known
results for scalar functions. As a particular example, it is shown that the
operators arising in continuous limits of coupled Kuramoto oscillators fail
everywhere to be Fr\'echet differentiability or locally compact.
