Let $p$ be a prime and $\mathbb{C}_{p}$ the completion of the algebraic closure of the $p$-adic number field $\mathbb{Q}_p$.

Let $M$ be a $\mathbb{Q}_p$-Banach space.

 We denote by $M\mathbin{\widehat{\otimes}_{\mathbb{Q}_p}}\mathbb{C}_{p}$ the complete tensor product over $\mathbb{Q}_p$.

Is the natural map $M\rightarrow M\mathbin{\widehat{\otimes}_{\mathbb{Q}_p}}\mathbb{C}_{p}$ an isometry ?