A norm-closed subspace \(\mathcal A\subseteq B(X,Y)\) is hyperreflexive if there is a constant \(C\) such that for all \(T\in B(X,Y)\) \[ \inf_{a\in{\mathcal A}}\| T-a\|\leq C\sup\| uTv\|, \] where the supremum is taken over all finite rank contractions \(u\) and \(v\) such that \(u\mathcal A v=0\); see \textit{V.\,Müller} and \textit{M.\,Ptak} [J.~Funct.\ Anal.\ 218, No.\,2, 395--408 (2005; Zbl 1074.47032)]. In the present paper, using normed operator ideals \((\mathcal B,\beta)\), a more general definition of \textit{\(\mathcal B\)-hyperreflexivity} is given by replacing all occurrences of \(\|\cdot\|\) in the definition above by the ideal norm \(\beta\). Also, considering only subsets \({\mathcal A}\) instead of subspaces, a modification leads to the definition of \textit{\(\mathcal B\)-Azoff--Shehada hyperreflexivity}. Examples of spaces having and failing \((\mathcal B,\beta)\)-hyperreflexivity are given. In the last section, this type of hyperreflexivity is used to construct operator spaces with prescribed sets of completely bounded maps.