  We begin a study of a pro-$p$ analogue of limit groups via extensions of
centralizers and call $\mathcal{L}$ this new class of pro-$p$ groups. We show
that the pro-$p$ groups of $\mathcal{L}$ have finite cohomological dimension,
type $FP_{\infty}$ and non-positive Euler characteristic. Among the group
theoretic properties it is proved that they are free-by-(torsion-free poly
-procyclic) and if non-abelian do not have a finitely generated non-trivial
normal subgroup of infinite index. Furthermore it is shown that every 2
generated pro-$p$ group in the class $\mathcal{L}$ is either free pro-$p$ or
abelian.
