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A study of the B$^+\to $J$/\psi\overline{\Lambda}$p decay using proton-proton collision data collected at $\sqrt{s}=$ 8 TeV by the CMS experiment at the LHC, corresponding to an integrated luminosity of 19.6 fb$^{-1}$, is presented. The ratio of branching fractions $\mathcal{B}$(B$^+ \to $J$/\psi\overline{\Lambda}$p )$/\mathcal{B}$(B$^\to $J$/\psi$K$^*$(892)$^+$) is measured to be (1.054 $\pm$ 0.057 (stat) $\pm$ 0.035 (syst) $\pm$ 0.011 ($\mathcal{B}$))%, where the last uncertainty reflects the uncertainties in the world-average branching fractions of $\overline{\Lambda}$ and K$^*$(892)$^+$ decays to reconstructed final states. The invariant mass distributions of the J$/\psi\overline{\Lambda}$, J$/\psi$p, and $\overline{\Lambda}$p systems produced in the B$^+\to $J$/\psi\overline{\Lambda}$p decay are investigated and found to be inconsistent with the pure phase space hypothesis. The analysis is extended by using a model-independent angular amplitude analysis, which shows that the inclusion of contributions from excited kaons decaying to the $\overline{\Lambda}$p system improves the description of the observed invariant mass distributions.