  An angular analysis of $B^0\to J/\psi K^+\pi^-$ decays is performed, using
proton-proton collision data corresponding to an integrated luminosity of
3~${\rm fb}^{-1}$ collected with the LHCb detector. The $m(K^+\pi^-)$ spectrum
is divided into fine bins. In each $m(K^+\pi^-)$ bin, the hypothesis that the
three-dimensional angular distribution can be described by structures induced
only by $K^\ast$ resonances is examined, making minimal assumptions about the
$K^+\pi^-$ system. The data reject the $K^\ast$-only hypothesis with a large
significance, implying the observation of exotic contributions in a
model-independent fashion. Inspection of the $m(J/\psi\pi^-)$ versus
$m(K^+\pi^-)$ plane suggests structures near $m(J/\psi\pi^-)=4200$~MeV and
4600~MeV.
