  We construct a non-trivial type of 1-step exceptional Bannai-Ito polynomials
which satisfy discrete orthogonality by using a generalized Darboux
transformation. In this generalization, the Darboux transformed Bannai-Ito
operator is directly obtained through an intertwining relation. Moreover, the
seed solution, which consists of a gauge factor and a polynomial part, plays an
important role in the construction of these 1-step exceptional Bannai-Ito
polynomials. And we show that there are 8 classes of gauge factors. We also
provide the eigenfunctions of the corresponding multiple-step exceptional
Bannai-Ito operator which can be expressed as a 3 x 3 determinant.
