A search for Higgs-boson pair production in the $b\bar{b}b\bar{b}$ final state is carried out with 3.2~fb$^{-1}$ of proton--proton collision data collected at $\sqrt{s} = 13$ TeV with the ATLAS detector. The data are consistent with the estimated background and are used to set upper limits on the production cross section of Higgs-boson pairs times branching ratio to $b\bar{b}b\bar{b}$ for both nonresonant and resonant production. In the case of resonant production of Kaluza--Klein gravitons within the Randall--Sundrum model, upper limits in the 24 to 91 fb range are obtained for masses between 600 and 3000 GeV, at the 95% confidence level. The production cross section times branching ratio for nonresonant Higgs-boson pairs is also constrained to be less than 1.22 pb, at the 95% confidence level.

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