The possible existence of a sign-changing gap symmetry in BaFe$_{2}$As$_{2}$-derived superconductors (SC) has been an exciting topic of research in the last few years. To further investigate this subject we combine Electron Spin Resonance (ESR) and pressure-dependent transport measurements to investigate magnetic pair-breaking effects on BaFe$_{1.9}M_{0.1}$As$_{2}$ ($M=$ Mn, Co, Cu, and Ni) single crystals. An ESR signal, indicative of the presence of localized magnetic moments, is observed only for $M=$ Cu and Mn compounds, which display very low SC transition temperature ($T_{c}$) and no SC, respectively. From the ESR analysis assuming the absence of bottleneck effects, the microscopic parameters are extracted to show that this reduction of $T_{c}$ cannot be accounted by the Abrikosov-Gorkov pair-breaking expression for a sign-preserving gap function. Our results reveal an unconventional spin- and pressure-dependent pair-breaking effect and impose strong constraints on the pairing symmetry of these materials.
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