A search for an invisibly decaying Higgs boson or dark matter candidates produced in association with a leptonically decaying $Z$ boson in proton-proton collisions at $\sqrt{s} =$ 13 TeV is presented. This search uses 36.1 fb$^{-1}$ of data collected by the ATLAS experiment at the Large Hadron Collider. No significant deviation from the background expectation is observed. Assuming the Standard Model $ZH$ production cross-section, an observed (expected) upper limit of 67% (39%) at the 95% confidence level is set on the branching ratio of invisible decays of the Higgs boson with mass $m_H =$ 125 GeV. The corresponding limits on the production cross-section of the $ZH$ process with the invisible Higgs decays are also presented. Furthermore, exclusion limits on the dark matter candidate and mediator masses are reported in the framework of simplified dark matter models.

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