  In-ice radio-detection is a promising technique to discover and characterize
ultra-high-energy (UHE) neutrinos, with energies above 100 PeV, adopted by
present - ARA, ARIANNA, and RNO-G - and planned - IceCube-Gen2. So far, their
ability to measure neutrino flavor has remained unexplored. We show and
quantify how the neutrino flavor can be measured with in-ice radio detectors
using two complementary detection channels. The first channel, sensitive to
$\nu_e$, identifies them via their charged-current interactions, whose radio
emission is elongated in time due to the Landau-Pomeranchuk-Migdal effect. The
second channel, sensitive to $\nu_\mu$ and $\nu_\tau$, identifies events made
up of multiple showers generated by the muons and taus they generate. We show
this in state-of-the-art forecasts geared at IceCube-Gen2, for representative
choices of the UHE neutrino flux. This newfound sensitivity could allow us to
infer the UHE neutrino flavor composition at their sources - and thus the
neutrino production mechanism - and to probe UHE neutrino physics.
