Absolutely. Third party garages don't void the warranty, the question here is the complete lack of proof that the car did the first service at all.
The previous owner appears to have been a private owner, and doesn't look like the vehicle was a lease... and after this ordeal started, the salesman claimed he was going to call the previous owner about this, so I'm expecting that either the previous owner didn't do the first year's service, or lost the paperwork, or just can't be bothered to look because it's not their problem anymore...