Consider a pseudo-spherical surface \(F^2 \subset E^3\), that is a surface with constant negative Gaussian curvature \(K=-1\). The classical Bianchi transformation provides a construction which gives a second pseudo-spherical surface \({\tilde F}^2\) which is regular in the generic case. If the transformed surface degenerates to a curve, the Bianchi transformation is said to be degenerate. For instance, when one starts with the classical Beltrami surface, that is rotating a tractrix around its asymptotic line, the resulting surface degenerates to the axis of rotation. In the present paper, the authors characterize all k-dimensional pseudo-spherical submanifolds with degenerate Bianchi transformation in the Euclidean space \(E^N\) for arbitrary \(k\geq 2\), \(N\geq 2k-1\). In a first step, generalized Beltrami submanifolds are constructed, using generalized tractrices. These submanifolds admit a degenerate Bianchi transformation. It turns out that, if the Bianchi transformation of a pseudo-spherical submanifold is degenerate, then it must be a generalized Beltrami submanifold.