  In this paper we continue our analysis on deformed canonical commutation
relations and on their related pseudo-bosons and bi-coherent states. In
particular, we show how to extend the original approach outside the Hilbert
space $\Lc^2(\mathbb{R})$, leaving untouched the possibility of defining
eigenstates of certain number-like operators, manifestly non self-adjoint, but
opening to the possibility that these states are not square-integrable. We also
extend this possibility to bi-coherent states, and we discuss in many details
an example based on a couple of superpotentials first introduced in
\cite{bag2010jmp}. The results deduced here belong to the same distributional
approach to pseudo-bosons first proposed in \cite{bag2020JPA}.
