We report the results of a search for the B→Y(4260)K, Y(4260)→J/ψπ+π- decays. This study is based on a data sample corresponding to an integrated luminosity of 711 fb-1, collected at the (4S) resonance with the Belle detector at the KEKB asymmetric-energy e+e- collider. We investigate the J/ψπ+π- invariant mass distribution in the range 4.0 to 4.6 GeV/c2 using both B+→J/ψπ+π-K+ and B0→J/ψπ+π-KS0 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→Y(4260)K decays, respectively, taking into account the systematic uncertainties. These correspond to upper limits on the product of branching fractions, B(B+→Y(4260)K+)×B(Y(4260)→J/ψπ+π-)<1.4×10-5 and B(B0→Y(4260)K0)×B(Y(4260)→J/ψπ+π-)<1.7×10-5 at the 90% confidence level.