The ANtarctic Impulsive Transient Antenna (ANITA) NASA long-duration balloon payload completed its fourth flight in December 2016, after 28 days of flight time. ANITA is sensitive to impulsive broadband radio emission from interactions of ultra-high-energy neutrinos in polar ice (Askaryan emission). We present the results of two separate blind analyses searching for signals from Askaryan emission in the data from the fourth flight of ANITA. The more sensitive analysis, with a better expected limit, has a background estimate of $0.64^{+0.69}_{-0.45}$ and an analysis efficiency of $82pm2%$. The second analysis has a background estimate of $0.34^{+0.66}_{-0.16}$ and an analysis efficiency of $71pm6%$. Each analysis found one event in the signal region, consistent with the background estimate for each analysis. The resulting limit further tightens the constraints on the diffuse flux of ultra-high-energy neutrinos at energies above $10^{19.5}$ eV.
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