OK, i don't see a reason to do it, but in any case, there is no way (that i know of) to refresh only the board, you can refresh the entire page (like reload), and you can do it by inject JS (if you are using Jira server and not cloud).
Does it answer your needs?
In cloud you can't to that, the only solution in cloud was the "announcement banner" but i just remembered that they removed it from the new experienced Jira Cloud.
There is an open ticket to Atlassian about it, but till then, there is no way to inject JS, unless you create your own plugin for cloud to do what you want
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