  In this work, we calculate the $CP$-averaged branching ratios and direct
$CP$-violating asymmetries of the quasi-two-body decays $B_{(s)}\to P
\rho^\prime(1450), P\rho^{\prime\prime}(1700)\to P \pi\pi$ by employing the
perturbative QCD (PQCD) factorization approach, where $P$ is a light
pseudoscalar meson $K, \pi, \eta$ and $\eta^{\prime}$. The considered decay
modes are studied in the quasi-two-body framework by parameterizing the
two-pion distribution amplitude $\Phi_{\pi\pi}^{\rm P}$. The $P$-wave time-like
form factor $F_{\pi}$ in the resonant regions associated with the
$\rho^\prime(1450)$ and $\rho^{\prime\prime}(1700)$ is estimated based on
available experimental data. The PQCD predictions for the $CP$-averaged
branching ratios of the decays $B_{(s)}\to P\rho^\prime(1450),
P\rho^{\prime\prime}(1700)\to P\pi\pi$ are in the order of $10^{-7} \sim
10^{-9}$. The branching ratios of the two-body decays $B_{(s)}\to
P\rho^\prime(1450), P\rho^{\prime\prime}(1700)$ are extracted from the
corresponding quasi-two-body decay modes. The whole pattern of the pion form
factor-squared $|F_\pi|^2$ measured by {\it BABAR} Collaboration could also be
understood based on our studies. Our PQCD predictions will be tested by the
precise data from the LHCb and the future Belle II experiments.
