In [EGNO15, Example 7.20.6], a family of examples is presented: if $ H $ is a unimodular non-semisimple finite-dimensional Hopf algebra, then $ H^* $ serves as a connected Frobenius algebra object in the category ${\rm Rep}(H \otimes H^{\rm cop})$.
For additional examples, refer to [LW23, Section 6], which discusses various rigid Frobenius algebras within non-semisimple modular tensor categories.
References
[EGNO15] Etingof, Pavel; Gelaki, Shlomo; Nikshych, Dmitri; Ostrik, Victor. Tensor Categories. Mathematical Surveys and Monographs, 205. American Mathematical Society, Providence, RI, 2015. xvi+343 pp.
[LW23] Laugwitz, Robert; Walton, Chelsea. Constructing non-semisimple modular categories with local modules. Comm. Math. Phys. 403 (2023), no. 3, 1363--1409.