  Let $\frakX$ be a smooth $p$-adic formal scheme over $\calO_C$ with rigid
generic fiber $X$. In this paper, we construct a new period sheaf
$\calO\widehat \bC_{\pd}^+$ on $X_{\proet}$ and use it to establish an integral
$p$-adic Simspon correspondence for small $\OXp$-representations on
$X_{\proet}$ and small Higgs bundles on $\frakX_{\et}$ which is compatible with
the works on rational level. In particular, for a small $\OXp$-representations
$\calL$ with induced Higgs bundle $(\calH,\theta_{\calH})$, we provide a
canonical morphism $\HIG(\calH,\theta_{\calH})\to\rR\nu_*\calL$ with a
uniformly bounded $p^{\infty}$-torsion cofiber. Finally, we shall use this
canonical map to study an analogue of Deligne--Illusie decomposition with
coefficients in small $\OXp$-representations.
