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In this paper we deal with a stationary non-degenerate $p-$Kirchhoff typeproblem with critical non-linearity and a subcritical parametrizedperturbation. We work on bounded domains of the Euclidean space, without anyrestriction on the dimension or on $p>0$. Variational methods will be utilizedin combination with an analysis of the fibering functions of the energyfunctional, in order to obtain ground state solutions, as well as Mountain Passsolutions, depending on the values of the parameter. A local analysis of theenergy functional will allow us to obtain non-trivial solutions even beyond theextremal parameter.

On a Perturbed Critical p-Kirchhoff-Type Problem