You can compute the persistent homology of any point cloud embedded in a metric space. In the real-world applications of persistent homology I've come across so far, the data points all have (naturally) linear dimensions, and so the data clouds embed into $\mathbf{R}^n$.
Are there any examples of researchers using persistent homology to effectively study data sets with non-linear data? Or where the metric they have to impose on the data is not just the induced metric from $\mathbf{R}^n$? I'm looking over some of the examples given in this answer now to see what they've done.