Yes, exactly as Telegram, for example
More than half of network connections are made from mobile devices (for example, almost 60% of Wikipedia visitors = mobile clients).
In Android and iOS, the client is not running if it is not the active window. The operating system may unload an inactive application at any time, but the client still able to receive messages via push, even if it is not running.
Desktop clients (Telegram, ICQ) have the same behavior: interacting with client = Online.