To give an exact answer to your question, you would need to advise exactly what type and form of silver or silver product is going to be exported, ie is it in the form of finished, or part finished jewellery products, bullion, bar, wire or other unmanufactured form, or scrap or ore?
As a guideline, there do not appear to be any export restrictions on sales to Bahrain of silver jewellery products (finished or part manufactured), nor on silver ore. There are, however restrictions on sales of silver, gold & precious stones & other similar products to Iran, and also restrictions on sales of luxury goods to Syria. If the goods are ultimately destined for those markets, even if you sell & ship them in good faith or ignorance to a client in Bahrain, you may find yourself in breach of export controls; therefore you should make efforts to satisfy yourself that this is not the case.
If you want to check this for any specific products, you can visit the following online UK / EU Customs tariff: https://www.gov.uk/trade-tariff/sections then follow the link to the item which corresponds to your product. When you find it, go to the bottom of the page, then click on the option to find out about country-specific export restrictions for that commodity. You could also contact the Export Control Office at the Department of Business, Innovations & Skills, at tel; 020 7215 4594 or by email : firstname.lastname@example.org .
This answer does not take into account any import restrctions in Bahrain, and you should also be aware of money laundering rstrictions, which govern any cash sales of jewellery products with a value of over £10,000.