Biological, soft fiber-reinforced composites, and more specifically tendon - which is aunidirectionally reinforced composite with continuous long fibers - follow a different paradigm from engineered composites. Instead of combining a ductile matrix with stiff and brittle fibers, collagen fibers in tendons exhibit significant ductility which allows for increased toughness, extensibility and resilience to cyclic loading. In this work, we aim to provide a fundamental mechanistic understanding of the response of soft composites with fiber plasticity. Specifically, the emergence of kinking instabilities in soft and biological composites is investigated, a phenomenon that has significant biomechanical implications. Uniquely, this kinking phenomenon does not require any macroscopic compressive loading in the direction of the fibers. To this end, we investigate the necessary features that could allow for a similar instability in a soft fiber-reinforced composite that shares some common features with tendon, namely a ductile and stiff fiber phase corresponding to collagen fibers, and a soft and elastic matrix phase corresponding to proteoglycans and elastin. We relate fiber plasticity in soft composites to the emergence of localization under non-monotonic uniaxial loading. We use homogenized models to characterize the macroscopic response of the composite, and then formulate a general loss of ellipticity criterion for an elastoplastic material subjected to finite deformations. We use this criterion to find the critical conditions under which localization occurs in the soft composite. Results show that plastic deformation of the fiber phase during tensile loading can lead to ellipticity breakdown during elastic unloading while, macroscopically, the material is still in tension, indicating the possible onset of an instability which may be related to fiber kinking that is observed in tendons.