Is it possible to partially format text in an <input> or a <textarea> box ?
When writting an email with Gmail, the content of the message can be formatted as in the following example:
I know that I can find existing JS libraries to write message like in Stack Overflow, but I would prefer something like Gmail (WYSIWYG).

It is not possible. All the WYSIWYG redactors uses the contentEditable attribute on
<div>(or another element) and the execCommand JS API to format the editor content.