Main Question. Is it consistent with ZFC that there exists a limit ordinal $\lambda$ and a cardinal number $\nu$ satisfying $$\beth_\lambda < \beth_\lambda^\nu < \beth_{\lambda+1}?$$
I am also interested in the following variant.
Secondary Question. Is it consistent with ZFC that there exists a limit ordinal $\lambda$ such that for some cardinal $\kappa < \beth_\lambda$, we can find a cardinal number $\nu$ satisfying the following? $$\beth_\lambda < \kappa^\nu < \beth_{\lambda+1}$$
Clearly neither statement is a theorem of ZFC, since GCH implies that the answers to both questions are "no."