In this work, we consider conforming finite element discretizations of arbitrary polynomial degree $p ≥ 1$ of the Poisson problem. We propose a multilevel a posteriori estimator of the algebraic error. We prove that this estimator is reliable and efficient (represents a two-sided bound of the error), with a constant independent of the degree $p$. We next design a multilevel iterative algebraic solver from our estimator and we show that this solver contracts the algebraic error on each iteration by a factor bounded independently of $p$. Actually, we show that these two results are equivalent. The $p$-robustness results rely on the work of Schöberl et al. [IMA J. Numer. Anal., 28 (2008), pp. 1-24] for one given mesh. We combine this with the design of an algebraic residual lifting constructed over a hierarchy of nested, unstructured simplicial meshes, in the spirit of Papež et al. [HAL Preprint 01662944, 2017]. This includes a global coarse-level lowest-order solve, with local higher-order contributions from the subsequent mesh levels. These higher-order contributions are given as solutions of mutually independent Dirichlet problems posed over patches of elements around vertices. This residual lifting is the core of our a posteriori estimator and determines the descent direction for the next iteration of our multilevel solver. Its construction can be seen as one geometric V-cycle multigrid step with zero pre- and one post-smoothing by damped additive Schwarz. Numerical tests are presented to illustrate the theoretical findings.