Following ideas of Caffarelli and Silvestre in [20], and using recent progress in hyperbolic fillings, we define fractional p-Laplacians (−∆ p ) θ with 0 < θ < 1 on any compact, doubling metric measure space (Z, d, ν), and prove existence, regularity and stability for the non-homogenous non-local equation (−∆ p ) θ u = f. These results, in turn, rest on the new existence, global Hölder regularity and stability theorems that we prove for the Neumann problem for p-Laplacians ∆ p , 1 < p < ∞, in bounded domains of measure metric spaces endowed with a doubling measure that supports a Poincaré inequality. Our work also includes as special cases much of the previous results by other authors in the Euclidean, Riemannian and Carnot group settings. Unlike other recent contributions in the metric measure spaces context, our work does not rely on the assumption that (Z, d, ν) supports a Poincaré inequality. * The research of N.S. and J.K. are partially funded by the NSF grants #DMS-1800161 and #DMS-2054960. The research of M.S. is partially funded by a supplemental grant from NSF #DMS-1800161. The research of L.C. is partly funded by NSF #DMS-195599. R.K. was supported by Academy of Finland, project 308063. The authors thank Mathav Murugan and Zhen-Qing Chen for illuminating discussion on jump processes and for pointing out the references [26,27,28] to us when N.S. and they were visiting at MSRI in Spring 2022, and Yannick Sire for pointing out the reference [22].