On metrizable non-Archimedean LF-spaces

It is known that no non-Archimedean LB-space (and no strict non-Archimedean LF-space) is metrizable. We show that there exist many metrizable (or even normable) non-Archimedean LF-spaces. We prove that every non-normable polar non-Archimedean Fréchet space (and every non-Archimedean Banach space with an infinite basis (xα)) contains a dense subspace which is an LF-space.