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Recent positron flux fraction measurements in cosmic-rays (CR) made by theAMS-02 satellite confirm and extends the evidence on the existence of a new(yet unknown) source of high energy positrons. To explain this excess, we usethe gravitino of bilinear R-parity violating SUSY models as a decaying DarkMatter candidate, as the source of those high energy particles. Being a longlived weak-interacting and spin 3/2 particle, it offers several particularitieswhich makes it an attractive candidate. We compute the electron, positron and$\gamma$-ray\ fluxes produced by each gravitino decay channel at the Earth.Combining the flux from the different decay modes we can fit AMS-02measurements of the positron fraction, as well as the electron and positronfluxes, with a gravitino mass in the range $1-2$ TeV and lifetimes of $\sim1.0-0.8\times 10^{26}$ s. . Then, we study the viability of these scenariosthrough their implications in $\gamma$-ray observations. We set limits on thegravitino lifetime using the Extragalactic $\gamma$-ray Background recentlyreported by the {\it Fermi}-LAT Collaboration and a state-of-the-art model ofits known contributors. These limits exclude the gravitino parameter spacewhich provides an acceptable explanation of the AMS-02 data. Therefore, weconclude that the gravitino of bilinear R-parity violating models is ruled outas the unique primary source of electrons and positrons needed to explain therise in the positron fraction.

Confronting recent AMS-02 positron fraction and Fermi-LAT Extragalactic Gamma-Ray Background measurements with gravitino dark matter