  We discuss the quantum-mechanical scattering of a massless scalar field on a
$\delta$-potential in a ghost-free theory and obtain analytic solutions for the
scattering coefficients. Due to the non-locality of the ghost-free theory the
transmission coefficient tends to unity for frequencies much larger than the
inverse scale of non-locality, even for infinitely strong potentials. At the
same time there exists a critical strength of the $\delta$-potential barrier
below which there is always a frequency that is totally reflected. These
scattering properties in ghost-free theories are quite generic and distinguish
them from local field theories. Moreover, we study quasi-normal states that are
present for the $\delta$-potential well. In the limit of vanishing
non-locality, we recover the standard results of local field theory.
