Can bathroom extractor fan work without switching light on?

If you want a fan that can be switched on and off at will, then the current wiring would need to be altered and a second switch added to control the fan independent of the light. Alternatively you could fit a fan that works on humidity or even use a passive infra red detector so it automatically senses when someone has entered the room. It all really depends on how much your budget is and what level of disturbance your prepared to accept to the current décor in the bathroom.

Building regulations recommend the bathroom extractor fan is controlled via the light switch, however there are a number of other solutions such as occupancy sensors or humidity sensors (integrated into fan) which offer an ideal solution for users who want automatic controllability.

There are many fans on the market, to answer the question in a simple format, yes you can get a fan that will run on its own with out a light. The one you have installed could be done in a way where an extra pull cord switch is installed so it can work on its own. If the one you have comes on and has an overrun ie stays on for a while after the light go off you can simply switch light on, then off straight away it will start the overrun which will be set at whatever time is allotted, which sometimes can be adjusted and range from anything from 1min to 30min depending on make and mod. If in doubt get a sparky to look and talk you through the options you have.