  A search for the rare decays $B_s^0 \to \pi^+\pi^-\mu^+\mu^-$ and $B^0 \to
\pi^+\pi^-\mu^+\mu^-$ is performed in a data set corresponding to an integrated
luminosity of 3.0 fb$^{-1}$ collected by the LHCb detector in proton-proton
collisions at centre-of-mass energies of 7 and 8 TeV. Decay candidates with
pion pairs that have invariant mass in the range 0.5-1.3 GeV/$c^2$ and with
muon pairs that do not originate from a resonance are considered. The first
observation of the decay $B_s^0 \to \pi^+\pi^-\mu^+\mu^-$ and the first
evidence of the decay $B^0 \to \pi^+\pi^-\mu^+\mu^-$ are obtained and the
branching fractions, restricted to the dipion-mass range considered, are
measured to be $\mathcal{B}(B_s^0 \to \pi^+\pi^-\mu^+\mu^-)=(8.6\pm 1.5\,({\rm
stat}) \pm 0.7\,({\rm syst})\pm 0.7\,({\rm norm}))\times 10^{-8}$ and
$\mathcal{B}(B^0 \to \pi^+\pi^-\mu^+\mu^-)=(2.11\pm 0.51\,({\rm stat}) \pm
0.15\,({\rm syst})\pm 0.16\,({\rm norm}) )\times 10^{-8}$, where the third
uncertainty is due to the branching fraction of the decay $B^0\to J/\psi(\to
\mu^+\mu^-)K^*(890)^0(\to K^+\pi^-)$, used as a normalisation.
