Jonathan Bardo

Note: This plugin only works with WordPress 4.2 or higher
This plugin will enable the site administrator to easily create an admin message present on all pages. The admin has the option to let the user dismiss the message for the duration of his session.

Admin User Message

This does just what I want. Thanks for making it. :)
It's definitely better than https://wordpress.org/plugins/admin-screen-messages
You might want to also tag it as:
https://wordpress.org/plugins/tags/admin-message



Perfect

Add message to users of your site. Choose wheter they can dismiss it or not.

Radek Mezulanik

You can write a main message, visible to everybody in dashboard widget. You can also write a bunch of messages in different dashboard widget, visible for all users with rights. 
Admin can read and write all the messages and setup user roles, who can read and write custom messages.

Zedna Custom Dashboard Messages

Works as expected

Allow admin to write messages on user dashboard.

Admin User Message vs Zedna Custom Dashboard Messages