We have used the Arecibo Telescope and the Green Bank Telescope to carry out a deep search for H{\sc i}~21\,cm emission from a large sample of Green Pea galaxies, yielding 19 detections, and 21 upper limits on the H{\sc i} mass. We obtain H{\sc i} masses of $\rm M_{HI} \approx (4-300) \times 10^8 \, \rm M_\odot$ for the detections, with a median H{\sc i} mass of $\approx 2.6 \times 10^9 \, \rm M_\odot$; for the non-detections, the median $3\sigma$ upper limit on the H{\sc i} mass is $\approx 5.5 \times 10^8 \, \rm M_\odot$. These are the first estimates of the atomic gas content of Green Pea galaxies. We find that the H{\sc i}-to-stellar mass ratio in Green Peas is consistent with trends identified in star-forming galaxies in the local Universe. However, the median H{\sc i} depletion timescale in Green Peas is $\approx 0.6$~Gyr, an order of magnitude lower than that obtained in local star-forming galaxies. This implies that Green Peas consume their atomic gas on very short timescales. A significant fraction of the Green Peas of our sample lie $\gtrsim 0.6$~dex ($2\sigma$) above the local $\rm M_{HI} - M_B$ relation, suggesting recent gas accretion. Further, $\approx 30$\% of the Green Peas are more than $\pm 2\sigma$ deviant from this relation, suggesting possible bimodality in the Green Pea population. We obtain a low H{\sc i}~21\,cm detection rate in the Green Peas with the highest O32~$\equiv$~[O{\sc iii}]$\lambda$5007/[O{\sc ii}]$\lambda$3727 luminosity ratios, O32~$> 10$, consistent with the high expected Lyman-continuum leakage from these galaxies.

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