In the beginning of the 7th chapter of the book "Spectral theory and analytic geometry over non-Archimedean fields" by Vladimir Berkovich one can find the phrase "...tensor product functor is exact on the category of Banach spaces...". He gave no clue how to prove it, but it is known that the same fact is not true for Archimedean Banach spaces. Is the statement correct, and how can it be proved?

**UPD**
I know that tensor product of complex Banach spaces is not left exact. I'm interested in the proof (or counterexample) for non-Archimedean Banach spaces.