Found the solution: this netlify.toml file will do the trick:
# The following redirect is intended for use with most SPAs that handle
# routing internally.
[[redirects]]
from = "/*"
to = "/index.html"
status = 200
[[headers]]
# Define which paths this specific [[headers]] block will cover.
for = "/*"
[headers.values]
Access-Control-Allow-Origin = "*"