  Let $\mathbb{K}$ be a non-Archimedean (complete) valued field satisfying
\begin{align*} \left|\sum_{j=1}^{n}\lambda_j^2\right|=\max_{1\leq j \leq
n}|\lambda_j|^2, \quad \forall \lambda_j \in \mathbb{K}, 1\leq j \leq n,
\forall n \in \mathbb{N}. \end{align*} For $d\in \mathbb{N}$, let
$\mathbb{K}^d$ be the standard $d$-dimensional non-Archimedean Hilbert space.
Let $m \in \mathbb{N}$ and $\text{Sym}^m(\mathbb{K}^d)$ be the non-Archimedean
Hilbert space of symmetric m-tensors. We prove the following result. If
$\{\tau_j\}_{j=1}^n$ is a collection in $\mathbb{K}^d$ satisfying $\langle
\tau_j, \tau_j\rangle =1$ for all $1\leq j \leq n$ and the operator
$\text{Sym}^m(\mathbb{K}^d)\ni x \mapsto \sum_{j=1}^n\langle x, \tau_j^{\otimes
m}\rangle \tau_j^{\otimes m} \in \text{Sym}^m(\mathbb{K}^d)$ is diagonalizable,
then \begin{align} (1) \quad \quad \quad \max_{1\leq j,k \leq n, j \neq
k}\{|n|, |\langle \tau_j, \tau_k\rangle|^{2m} \}\geq \frac{|n|^2}{\left|{d+m-1
\choose m}\right| }. \end{align} We call Inequality (1) as the non-Archimedean
version of Welch bounds obtained by Welch [\textit{IEEE Transactions on
Information Theory, 1974}]. We formulate non-Archimedean Zauner conjecture.
