  For an algebraic number $\alpha$, let $h(\alpha)$ denote its logarithmic Weil
height. In 2002, Bombieri and Zannier obtained lower bounds for Weil height for
totally $p$-adic extensions of $\mathbb{Q}$, i.e., infinite Galois extension
over $\mathbb{Q}$ with finite local degree over a rational prime $p$. In this
paper, we formulate and prove the non-Galois analogue of this result. In
particular, it leads to a $p$-adic equidistribution result i.e., if a positive
proportion of conjugates of $\alpha\in\overline{\mathbb{Q}}$ lie in a finite
local extension $K_{v}/\mathbb{Q}_p$, then we obtain absolute lower bound for
$h(\alpha)$. We also undertake the analoguous problem for canonical height of
an elliptic curve $E/K$ and obtain explicit lower bounds for them on certain
non-Galois infinite extensions.
