We construct a noncommutative, separably represented, type I and approximately finite dimensional C^* -algebra such that its multiplier algebra is equal to its unitization. This algebra is an essential extension of the algebra \mathcal K(\ell_2(\mathfrak{c})) of compact operators on a nonseparable Hilbert space by the algebra \mathcal K(\ell_2) of compact operators on a separable Hilbert space, where \mathfrak{c} denotes the cardinality of continuum. Although both \mathcal K(\ell_2(\mathfrak{c})) and \mathcal K(\ell_2) are stable, our algebra is not. This sheds light on the permanence properties of the stability in the nonseparable setting. Namely, unlike in the separable case, an extension of a nonseparable C^* -algebra by \mathcal K(\ell_2) does not have to be stable. Our construction can be considered as a noncommutative version of Mrówka’s \Psi -space; a space whose one point compactification is equal to its Cech–Stone compactification and is induced by a special uncountable family of almost disjoint subsets of \mathbb{N} .