Your gun is a BSA Standard in No.1 bore (.177) with an A prefix. It was manufacture in the period 1936-1939, and was the follow on model from the BSA standard No.1 bore with the L prefix, which was made 1919-1935 ish.
They made about 30,000 of the L prefix guns but under 1000 of the A prefix guns, so relatively speaking they are a rareish model .

Hope that helps

Lakey