  We investigate the existence of double copy structure, or the lack thereof,
in higher derivative operators for Nambu-Goldstone bosons. At the leading
${\cal O}(p^2)$, tree amplitudes of Nambu-Goldstone bosons in the adjoint
representation can be (trivially) expressed as the double copy of itself and
the cubic bi-adjoint scalar theory, through the Kawai-Lewellen-Tye bilinear
kernel. At the next-to-leading ${\cal O}(p^4)$ there exist four operators in
general, among which we identify one operator whose amplitudes exhibit the
flavor-kinematics duality and can be written as the double copy of ${\cal
O}(p^2)$ Nambu-Goldstone amplitudes and the Yang-Mills+$\phi^3$ theory,
involving both gluons and gauged cubic bi-adjoint scalars. The specific
operator turns out to coincide with the scalar ${\cal O}(p^4)$ operator in the
so-called extended Dirac-Born-Infeld theory, for which the aforementioned
double copy relation holds more generally.
