  We study the asymptotical behavior of the $p$-adic singular Fourier integrals
$$ J_{\pi_{\alpha},m;\phi}(t) =\bigl< f_{\pi_{\alpha};m}(x)\chi_p(xt),
\phi(x)\bigr> =F\big[f_{\pi_{\alpha};m}\phi\big](t), \quad |t|_p \to \infty,
\quad t\in \bQ_p, $$ where $f_{\pi_{\alpha};m}\in {\cD}'(\bQ_p)$ is a {\em
quasi associated homogeneous} distribution (generalized function) of degree
$\pi_{\alpha}(x)=|x|_p^{\alpha-1}\pi_1(x)$ and order $m$, $\pi_{\alpha}(x)$,
$\pi_1(x)$, and $\chi_p(x)$ are a multiplicative, a normed multiplicative, and
an additive characters of the field $\bQ_p$ of $p$-adic numbers, respectively,
$\phi \in {\cD}(\bQ_p)$ is a test function, $m=0,1,2...$, $\alpha\in \bC$. If
$Re\alpha>0$ the constructed asymptotics constitute a $p$-adic version of the
well known Erd\'elyi lemma. Theorems which give asymptotic expansions of
singular Fourier integrals are the Abelian type theorems. In contrast to the
real case, all constructed asymptotics have the {\it stabilization} property.
