This is a question about terminology and should be taken lightly.

The expression *local field* is used in at least three different senses :

1) For a locally compact totally disconnected field. These are the finite extensions of $\mathbb{Q}_p$ or of $\mathbb{F}_p((T))$, where $p$ is a prime number and $T$ is an indeterminate.

2) For a field complete with respect to a discrete valuation whose residue field is merely perfect, not necessarily finite as in 1). This is how Fontaine uses the expression.

3) For locally compact fields which are not discrete. These are the fields in 1), but also $\mathbb{R}$ and $\mathbb{C}$ in addition. People who adopt this definition refer to the fields in 1) as non-Archimedean local fields.

(Nobody insists that a local field is a local ring which happens to be a field, but the "logic" is impeccable.)

**Question.** Would *locally profinite field* be a good piece of terminology for the fields in 1) ?

This would certainly avoid the confusion with the other fields in 2) and 3).

The expression *locally profinite group* is already in use (for example in the book Bushnell-Henniart). The additive and the multiplicative groups of a locally profinite field would be locally profinite groups in their sense.