I have installed the modules below, but still can not set the permission for editing a block body with an specific role. Does someone know how to solve this issue?
Both of those modules look more like they are meant to simply display different blocks/content depending on user role. You may be better off utilising Nodes In Block. Create a content type, or use one of your existing content types and create a node that has the required block content. Lastly, allow certain roles permissions to edit that particular node. That will in turn update the block content!