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Q1. Let $K$ be a local field with valuation $v$. Let us call $K'\subset K$ a nice local subfield if it is complete with respect to the induced from $K$ valuation. By local subfield I will mean a subfield $K'\subset K$ which is complete with respect to some valuation $v'$, possibly different from the one induced from $K$. Is it true that any local subfield can be embedded as a nice local subfield? In particular, I'm interested in the case when $K$ is a higher local field.

Q2. Is it true that any local subfield of a higher local field is a higher local field? Is it true in the case of nice local fields?

EDIT: As I answered below, the answer to both question is no. But of course it is true for local fields of dimension one.

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The answer to both questions is no, even in the case of higher local fields.

Q1. Let us take $K = \mathbb Q_p((T))$ and $K' = \mathbb Q_p$. Then $K'$ is a local subfield of $K$ but there is only one embedding of $K'$ in $K$, and as a subfield of $K$, $K'$ is not a nice local subfield.

Q2. Take again $K=\mathbb Q_p((T))$. Then $K'=\mathbb Q((T))$ is a nice local subfield but it is not a higher local field.

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