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.

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*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.