@davegoodman offered a good solution, and the only one I know of: update the repeater datasheet (the values for the columns in your repeater rows) in order to make them persistent. So, in your example file you have a “default” and “edit” mode for each row. When the “save icon” is clicked, rather than setting the text on widgets in the default state directly, you should use an Update Rows action to update the datasheet values. This will cause the table to reload, showing the updated values. Because the data is in the repeater datasheet and not just in the (repeated) widgets, it will be retained when you change repeater-pages, hide/unhide rows, update other rows, etc.
it is important to note that as soon as you navigate away from your (web prototype) page in the browser, all this data will be lost. Each time to load a prototype page, its default state will be created. If you need to track what data is in your repeater across different prototype pages, then you will need to devise a scheme to represent it in global variable(s). Take a look at this thread for more info: