  A search for $B^{0}_{s}\to\tau^{\pm}\mu^{\mp}$ and
$B^{0}\to\tau^{\pm}\mu^{\mp}$ decays is performed using data corresponding to
an integrated luminosity of 3 fb$^{-1}$ of proton-proton collisions, recorded
with the LHCb detector in 2011 and 2012. For this search, the $\tau$ lepton is
reconstructed in the $\tau^{-}\to\pi^{-}\pi^{+}\pi^{-}\nu_{\tau}$ channel. No
significant signal is observed. Assuming no contribution from
$B^{0}\to\tau^{\pm}\mu^{\mp}$ decays, an upper limit is set on the
$B^{0}_{s}\to\tau^{\pm}\mu^{\mp}$ branching fraction of $\mathcal{B}\left(
B^{0}_{s}\to\tau^{\pm}\mu^{\mp}\right) < 4.2\times 10^{-5}$ at $95\%$
confidence level. If instead no contribution from
$B^{0}_{s}\to\tau^{\pm}\mu^{\mp}$ decays is assumed, a limit of
$\mathcal{B}\left( B^{0}\to\tau^{\pm}\mu^{\mp}\right) < 1.4\times 10^{-5}$ is
obtained at $95\%$ confidence level. These are the first limit on
$\mathcal{B}\left( B^{0}_{s}\to\tau^{\pm}\mu^{\mp}\right)$ and the world's best
limit on $\mathcal{B}\left( B^{0}\to\tau^{\pm}\mu^{\mp}\right)$.
