We investigate $B_s\pi^+$ scattering in s-wave using lattice QCD in order to search for an exotic resonance X(5568) with flavor $\bar b s \bar d u$; such a state was recently reported by D0 but was not seen by LHCb. If X(5568) with $J^P=0^+$ exists, it can strongly decay only to $B_s\pi^+$ and lies significantly below all other thresholds, which makes a lattice search for X(5568) cleaner and simpler than for other exotic candidates... Both an elastic resonance in $B_s\pi^+$ as well as a deeply bound $B^+\bar K^0$ would lead to distinct signatures in the energies of lattice eigenstates, which are not seen in our simulation. We therefore do not find a candidate for X(5568) with $J^P=0^+$ in agreement with the recent LHCb result. The extracted $B_s\pi^+$ scattering length is compatible with zero within the error. (read more)