This repo contains a small demo in-app notifications for GTK. In-app notifications are part of GTK, but there's no special widget. Instead it can be created by combining existing widgets and applying style classes.
When you look at applications like GNOME Files and Geary, it shows a nice small notification overlay at the top of the window every now and then.
There is no special widget for such a notification bar. When I searched for this notification bar I found exactly one match: a post on Stack Overflow. Where else.
That's not a lot. Geary is not the only application to implement one.
The basic idea is to use a Gtk.Overlay.
Add the main content to the base layer and add a second layer with a Gtk.Revealer.
To this revealer, you add a Gtk.Box,
or a Gtk.Frame, and apply a class app-notification to it. Add a label and a button and you're done. The icons used are "symbolic" icons.