In non-Archimedean locally compact abelian groups such as the p-adic numbers, every compact open set admits a Riesz basis of characters, but some bounded open sets exclude all such bases.
Christensen, An introduction to frames and Riesz bases, Applied and Numerical Harmonic Analysis
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Exponential Riesz bases in non-Archimedean locally compact Abelian groups
In non-Archimedean locally compact abelian groups such as the p-adic numbers, every compact open set admits a Riesz basis of characters, but some bounded open sets exclude all such bases.