  We study $B\to\pi\pi$ form factors using QCD light-cone sum rules with
$B$-meson distribution amplitudes. These form factors describe the semileptonic
decay $B\to \pi\pi \ell\bar{\nu}_{\ell}$, and constitute an essential input in
$B\to \pi\pi \ell^+\ell^-$ and $B\to \pi\pi\pi$ decays. We employ the
correlation functions where a dipion isospin-one state is interpolated by the
vector light-quark current. We obtain sum rules where convolutions of the
$P$-wave $\bar{B}^0\to \pi^+\pi^0$ form factors with the time-like pion vector
form factor are related to universal $B$-meson distribution amplitudes. These
sum rules are valid in the kinematic regime where the dipion state has a large
energy and a low invariant mass, and reproduce analytically the known
light-cone sum rules for $B\to \rho$ form factors in the limit of
$\rho$-dominance with zero width, thus providing a systematics for
so-far-unaccounted corrections to $B\to\rho$ transitions. Using data for the
pion vector form factor, we estimate finite width-effects and the contribution
of excited $\rho$-resonances to the $B\to\pi\pi$ form factors. We find that
these contributions amount up to $\sim 20\%$ in the small dipion mass region
where they can be effectively regarded as a nonresonant ($P$-wave) background
to the $B\to\rho$ transition.
