Many $p$-adic odd zeta values are irrational

For any prime $p$ and $\epsilon >0$ we prove that for any sufficiently large positive odd integer $s$ at least $(c_p-\epsilon )\sqrt{\frac{s}{\log s}}$ of the $p$-adic zeta values ${\zeta }_p(3),{\zeta }_p(5),\dots ,{\zeta }_p(s)$ are irrational. The constant $c_p$ is positive and does only depend on $p$. This result establishes a $p$-adic version of the elimination technique used by Fischler–Sprang–Zudilin and Lai–Yu to prove a similar result on classical zeta values. The main difficulty consists in proving the non-vanishing of the resulting linear forms. We overcome this problem by using a new irrationality criterion.