  Single crystals of CaFe$_2$(As$_{1-x}$P$_x$)$_2$ are found to exhibit the
tetragonal (T) to collapsed-tetragonal (cT) transition at $T_{\rm cT}
\lesssim100$\,K for $x>0.05$. The c-axis shrinks by $\sim9$% below $T_{\rm
cT}$, which substantially diminishes the interband nesting between the hole and
electron bands. In sharp contrast to the superconducting T phase of
$A$Fe$_2$(As$_{1-x}$P$_x$)$_2$ ($A=$ Ba, Sr), where the anomalous non-Fermi
liquid transport properties are observed, the resistivity, Hall coefficient,
and magnetoresistance data in the Ca-based system all indicate that the
standard Fermi liquid behaviors are recovered abruptly below $T_{\rm cT}$, and
the superconductivity disappears completely. The intimate link between the
superconductivity and the non-Fermi liquid transport enlightens the essential
role of interband-associated fluctuation effects in Fe-pnictides.
