I've had a Dynamic Tempo chainless bike for about 12 years, but it finally broke down. It looks like Dynamic Bikes is no longer in business (or at least the web page for them is not - transfers to Bike share). Is there any chainless bikes sold in USA? I have seen some in other countries.
I am guessing you're thinking of similar shaft-driven bicycles when you say "chainless", and those are indeed rare (although CeramicSpeed is working on a shaft-driven concept). However, if you're looking for anything that doesn't have a chain, belt drive bikes are by far the most popular option, and are widely available from many different manufacturers including Specialized, Cannondale, and (previously, at least) Trek. Ask at your local bike shop and the odds are very high that they'll either have at least one belt drive option in stock or will be able to order one for you fairly easily.
Dynamic Bicycles seems to be in business in the UK. http://www.dynamicbicyclesuk.co.uk.
If your problem is not with the shaft system, the rest of the bikes appear to use standard components and can be dealt with by any competent bike mechanic.
If the issue is with the shaft, the company may be able to help you if you contact them.