No. DAB digital radio is a terrestrial technology using land-based transmitters. You can only listen to stations when you are within their transmission range.
If I take my DAB digital radio abroad, will I be able to pick up foreign DAB services?
Yes. In the UK, DAB services are broadcast on a band of spectrum called Band III. Many European countries also use Band III, although some use a different band of spectrum called L Band. Some countries, such as Germany, use Band III in one part of the country and L Band in another.
If the receiver you are using is designed only to pick up Band III, then it will work in other countries using Band III and you will be able to listen to local DAB services. If it is a dual-band receiver (that is, designed to work on either Band III or L Band) then it will pick up any DAB stations being broadcast on either Band III or L Band abroad.
If you specifically want a receiver that will work both in the UK and Europe, then make sure before you buy that it is a dual band receiver.