© 2014, Springer Basel. Let $$p>1$$p>1 and let $$\psi $$ψ be a structural set in $$\mathbb H$$H. This paper presents a seminormed $$\psi $$ψ-hyperholomorphic Besov space which satisfies the conformal covariant property preserving the “$$\psi $$ψ structure”. This is an analogue of what happens in the well-known one-dimensional complex Besov space, see Danikas (Function spaces and complex analysis, Joensuu 1997 (Ilomantsi) 935, 1999), Zhu (Operator theory in function spaces, 1990). These computations presented here are important complements to some results given in Cnops and Delanghe (Appl Anal 73:45–64, 1999) and Gürlebeck et al. (Complex Var Theory Appl 39:115–135, 1999) concerning to the theory of quaternionic right-linear space of $$\psi $$ψ-hyperholomorphic functions, see Shapiro and Vasilevski (Complex Var Theory Appl 27:17–46, 1995) and Shapiro and Vasilevski (Complex Var Theory Appl 27:67–96, 1995).