We report the results of a search for the $B\to Y\left(4260\right)K$, $Y\left(4260\right)\to J/\psi {\pi }^{+}{\pi }^{-}$ decays. This study is based on a data sample corresponding to an integrated luminosity of $711{\mathrm{fb}}^{-1}$, collected at the $\Upsilon \left(4S\right)$ resonance with the Belle detector at the KEKB asymmetric-energy $e^{+}{e}^{-}$ collider. We investigate the $J/\psi {\pi }^{+}{\pi }^{-}$ invariant mass distribution in the range 4.0 to $4.6\mathrm{GeV}/c^2$ using both $B^{+}\to J/\psi {\pi }^{+}{\pi }^{-}{K}^{+}$ and $B^0\to J/\psi {\pi }^{+}{\pi }^{-}{K}_S^0$ decays. We find excesses of events above the background levels, with significances of 2.1 and 0.9 standard deviations for charged and neutral $B\to Y\left(4260\right)K$ decays, respectively, taking into account the systematic uncertainties. These correspond to upper limits on the product of branching fractions, $B(B^{+}\to Y(4260\left)K^{+}\right)\times B\left(Y\right(4260)\to J/\psi {\pi }^{+}{\pi }^{-})<1.4\times {10}^{-5}$ and $B(B^0\to Y(4260\left)K^0\right)\times B\left(Y\right(4260)\to J/\psi {\pi }^{+}{\pi }^{-})<1.7\times {10}^{-5}$ at the 90% confidence level.