Raspberry leaves have orange rust

We’ve just discovered that our raspberry leaves are breaking out in bright orange, slightly-raised spots on the lower parts of the leaves. Orange dust falls from the pustules. Is this rust? How can we control it?

This is undoubtedly rust. Raspberries are quite commonly affected, especially during changeable weather and when the leaves are young and tender.
Raspberry rust seldom proves fatal, but if the disease spreads too much, productivity might be reduced. Your canes will benefit from being cleaned up.
The orange dust you mention is made up of fungal spores, so removing any leaves which have pustules will help to reduce the spread of the disease to clean foliage. Compost or dispose of them.
You could also spray your canes, using an approved fungicide. Take care to follow the instructions to the letter.