I would like to save files without exiting the INSERT mode. Is it possible? How?
Yes, just use the
<C-O> keystroke to temporarily leave insert mode for a single command. You can then run
:w<CR> to write the file and you'll be back to insert mode.
You can use that to create a mapping in insert mode and assign that to a key combination such as
inoremap <F3> <C-O>:w<CR>
Avoid a keystroke such as
<C-S> ("S" for "Save") since that specific keystroke can be a problem in many terminals or sometimes through remote connections...
<C-S> is the code for XOFF, which historically has been used to pause a terminal and it's still a valid terminal command in most modern terminals.
If you use a key such as
<F3>, you might want to also map it in normal mode (without the
<C-O> in that case) and visual mode, etc. so that it works regardless of which mode you're currently on.