Wise Chat Pro is an advanced and highly popular chat plugin for WordPress. It requires no additional server, supports multiple chat rooms, bad words filtering, pictures / files posting and detailed appearance settings. The plugin is well designed for mobile devices. It has single-sign-on feature, provides easy localization for every language, moderation, backups, opening hours control and much more.

Demo: https://kaine.pl/projects/wp-plugins/wise-chat-pro/