In this work, the notion of function of bounded variation in 2-normed spaces was established (Definition 4.2), the set of functions of bounded $(2,k)$-variation was endowed with a norm (Theorem 4.5), and it was proved that such a set is a Banach space (Theorem 4.6). In addition, the fundamental properties of the functions of bounded $(2,k)$-variation in the formalism of 2-Hilbert and 2-normed spaces were studied (see Theorems 4.1, 4.3, 4.4). Also, it was shown how to endow a 2-normed space with a function of bounded $(2,k)$-variation from a classical Hilbert space (Proposition 4.1). A series of examples and counterexamples are presented that enrich the results obtained in this work (4.1 and 4.2).