Exceptional points are spectral singularities of open systems, where several eigenvalues and eigenvectors coalesce. In photonics, they are associated to remarkable phenomena, such as unidirectional scattering, enhanced sensing or chiral mode conversion. In this work, we study scattering of electromagnetic waves by a single dielectric nanoparticle and observe the appearance of exceptional points in its eigenvalue spectrum. Their existence is linked to breaking the mirror symmetry of the particle. Remarkably, they mark the onset from weak to strong coupling of the resonant modes. We discuss in detail the example of the electric and magnetic dipole modes supported by a silicon nanoparticle. We argue that any two modes of a resonant dielectric nanoparticle can merge to create an exceptional point, provided their resonant frequencies cross as functions of a parameter such as, e.g., aspect ratio, and their field distributions have opposite signs after a reflection in the transverse plane of the structure. The strongly coupled modes radiate as a mixture of electric and magnetic dipoles resulting in an intense bianisotropic response, being easily controlled by symmetry-breaking perturbations. We also study the effect of a dielectric substrate and demonstrate that the latter provides an additional mechanism to tune the position of exceptional points in the parameter space. Finally, we discuss applications of bianisotropic EPs, including their use for refractive-index sensing. © 2024 authors. Published by the American Physical Society. Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.