Squeezing light through nanometre-wide gaps in metals can lead to extreme field enhancements, nonlocal electromagnetic effects and light-induced electron tunnelling. This intriguing regime, however, has not been readily accessible to experimentalists because of the lack of reliable technology to fabricate uniform nanogaps with atomic-scale resolution and high throughput. Here we introduce a new patterning technology based on atomic layer deposition and simple adhesive-tape-based planarization. Using this method, we create vertically oriented gaps in opaque metal films along the entire contour of a millimetre-sized pattern, with gap widths as narrow as 9.9 Å, and pack 150,000 such devices on a 4-inch wafer. Electromagnetic waves pass exclusively through the nanogaps, enabling background-free transmission measurements. We observe resonant transmission of near-infrared waves through 1.1-nm-wide gaps (λ/1,295) and measure an effective refractive index of 17.8. We also observe resonant transmission of millimetre waves through 1.1-nm-wide gaps (λ/4,000,000) and infer an unprecedented field enhancement factor of 25,000.