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In this paper, we compute certain $p$-adic zeta integrals appearing in thedoubling method of Garrett and Piatetski-Shapiro-Rallis for unitary groups.Using structure theorems in the author's work arXiv:2310.09110 for$P$-(anti-)ordinary automorphic representations involving Bushnell-Kutzkotypes, we associate local Siegel-Weil sections at $p$ to such Bushnell-Kutzkotypes. Then, fixing compatible choices of $P$-anti-ordinary vectors, we findexplicit formulae relating the corresponding $p$-adic zeta integral to modified$p$-Euler factors and volumes of $P$-Iwahoric subgroups. Our results extend theones of Eischen, Harris, Li and Skinner for the ordinary setting by allowingautomorphic representations with nontrivial supercuspidal support at $p$.

$p$-adic zeta integrals on unitary groups via Bushnell-Kutzko types