The NHS contact-tracing app has been downloaded more than 55,000 times since being launched on the Isle of Wight.
Letters with instructions were sent to all the island's 80,000 households after the app went live on Tuesday.
Island MP Bob Seely said residents were "leading the way for the country". If the trial is successful, the app is due to be rolled out nationally.
However, the app does not record location so it is not known if all downloads were on the island.
Mr Seely told the Local Democracy Reporting Service (LDRS): "We shouldn't be complacent.
"We need to make sure that everyone who can download it, does so, but those who cannot need to be sending their feedback... we can still do better."
How does the app work?
The new app - published on Apple and Google's app stores - works by using a Bluetooth connection.
It records when two people who have the app are within a certain distance of each other for longer than a specified amount of time.
If one of those people later reports having symptoms, all the other app users they came into significant contact with over recent days will be alerted and, if judged necessary, told to self-isolate.
"The exact advice on what you should do will depend on the evolving context and approach," the NHS has said.
Dr Geraint Lewis, NHS England's chief data officer who has been overseeing the project, said the response to the app on the island had been "overwhelming".
He said the app did not track location, only how far away a smartphone had been from another phone with the app installed, so it was impossible to know if all the downloads had been on the island.
"But the profile of how and when people started downloading gives me confidence that the vast, vast majority of what we are seeing is islanders," he added.
Dr Lewis said the app did not work on some older phones and encouraged people to update their operating systems and ensure they have Bluetooth technology.
Concerns have previously been raised over privacy and one island resident said he "won't be a guinea pig" for the project.
The government said the app had been designed with privacy at "front of mind".