  Using a data sample corresponding to an integrated luminosity of 10.9
fb$^{-1}$ collected at center-of-mass energies from 4.16 to 4.34 GeV with the
BESIII detector, we search for the decay $\chi_{c1}(3872) \to
\pi^{+}\pi^{-}\chi_{c1}$ in the radiative production $e^{+}e^{-} \to \gamma
\chi_{c1}(3872)$. No significant signal is observed, and the ratio for the
branching fraction of $\chi_{c1}(3872) \to \pi^{+}\pi^{-}\chi_{c1}$ to
$\chi_{c1}(3872) \to \pi^{+}\pi^{-}J/\psi$ is measured as
$\mathcal{R}\equiv\frac{\mathcal{B}[\chi_{c1}(3872) \to
\pi^{+}\pi^{-}\chi_{c1}]}{\mathcal{B}[\chi_{c1}(3872) \to \pi^{+}\pi^{-}
J/\psi]}<0.18$ at 90$\%$ confidence level. The upper limit on the product of
the cross section $\sigma[e^{+}e^{-}\to\gamma\chi_{c1}(3872)]$ and the
branching fraction $\mathcal{B}[\chi_{c1}(3872)\to\pi^{+}\pi^{-}\chi_{c1}]$ at
each center-of-mass energy is also given. These measurements favor the
non-conventional charmonium nature of the $\chi_{c1}(3872)$ state.
