They also produced their own bikes, like the Alpha highwheeler. In 1884 they produced one of the first models of the safety bicycle. 1500 of this BSA were made - click this link for a picture.   

BSA built bicycles until 1887 but remained an important supplier of bicycle parts, used all over the world.

The bike you see in the pictures is a very interesting example, seen in the Velorama museum. It has a lot of 'modern' features, like the hollow forks and the ball bearings. But this way of spoke adjustment was very oldfashioned by 1885: direct spokes were considered a better solution. 125 years later I can't see why ... 

Apart from that it misses a brake: not even the remains of it. Someone told me this was standard with BSA ordinaries: they were sold without brake. 

Characteristic is the way the handlebars are fixed to the head: with a giant nut. Simple, yet effective.