If the test succeeds on the island, which with a population of around 22,000 is one of the smallest in the Canary archipelago popular with tourists, it may be rolled out nationwide.
Authorities will give users randomly generated codes, some of which will show they are positive for COVID-19 for the purposes of the simulation, to record contacts between people using Bluetooth technology and alert them if they have been in contact with one of the "positive" cases.
Organisers hope 3,000 people will download the app, allowing them to inject 300 simulated positive diagnoses into the system to mimic a 10% infection rate.
Spain is trying a tool based on a standard developed by Apple and Google which has already been used by countries including Germany. It holds data on devices to preserve privacy - one of the main concerns as similar technology is rolled out worldwide.
After downloading the app, 34-year-old teacher Leticia Castilla said some people she knew were worried about privacy, but she was not."I don't think they will follow me. It's private and you don't give your name or surname."
Joel Serafin, a 46 year-old forestry technician who volunteered to help people download the app at a stand in the centre of San Sebastian, La Gomera's main town, said people were less concerned about data privacy than he had expected.
"Of every 10 people who come to me, perhaps only one asks about the privacy issue, others don't care," he said.
Spain's infection and death rate has drastically slowed since peaking in April. The first known case was identified on La Gomera in January, but the Canaries have since recorded just 2,436 of the country's total 250,545 cases.
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I want the authorities to know less about me, not more. So no app for me. I will charge my old Nokia 3310 again, just in case they make is compulsory.