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Let $\mathcal{O}_E$ be a complete discrete valuation ring and $R$ be aperfect ring in characteristic $p$, we also assume $R$ is a complete valuationring whose valuation group is of rank one and non-discrete, we prove the Krulldimension of the ring $W_{\mathcal{O}_E}(R)$ of $\mathcal{O}_E$-Witt vectorsover $R$ is at least the cardinality of the continuum.

$\mathbf{A}_{\text {inf}}$ has uncountable Krull dimension