The authors consider ultrametric spaces with a metric \(| xy| \) satisfying the strong triangle inequality \(| ab| \leq\max(| ac| ,| cd| )\) for all \(c\). They construct measures in ultrametric spaces and introduce bases in quadratically integrable (complex valued) function spaces. They introduce ultrametric change of variables as specific surjective maps for these spaces onto new orthonormal bases in \(L^2({\mathbb R}_+)\). The image of an ultrametric wavelet basis is called a nonhomogeneous wavelet basis in \(L^2({\mathbb R}_+)\), where nonhomogeneity means that, unlike in the case of the usual wavelet bases, vectors of a nonhomogeneous wavelet basis cannot be constructed using shifts and dilations of a fixed wavelet. Pseudodifferential operators acting on ultrametric spaces are shown to be diagonal in the bases of ultrametric wavelet bases. Applications to \(p\)-adic mathematical physics are mentioned. The above ultrametric spaces are a generalization of the space of \(p\)-adic numbers found in \textit{J. J. Benedetto} and \textit{R. L. Benedetto} [J. Geom. Anal. 14, No.~3, 423--456 (2004; Zbl 1114.42015)] where a \(p\)-adic wavelet basis in the space of quadratically integrable functions with \(p\)-adic argument was introduced and the natural map of \(p\)-adic numbers onto positive real numbers maps this basis onto the wavelet basis generated by the Haar wavelet.