Andi.
The Gambia Experience have just replied with the definitive answer. When the British were looking for a suitable place to ship the newly captured slaves to the US they found the Gambia river and that is actually all that the required. They then plonked the British flag on the river bank and declared (as you do) that this is now ours and British.
The upshot is that The River Gambia was the place know as 'The' and subsequently the Country became known as The Gambia.
The few other countries with 'the' titles have a similar explanation.