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We show that there exists a non-archimedean Fréchet-Montel space 
$W$ with a basis and with a continuous norm such that any non-archimedean Fréchet space of countable type is isomorphic to a quotient of $W$. We also prove that any non-archimedean nuclear Fréchet space is isomorphic to a quotient of some non-archimedean nuclear Fréchet space with a basis and with a continuous norm.
