Dark matter particles with Planck-scale mass ($simeq10^{19}text{GeV}/c^2$) arise in well-motivated theories and could be produced by several cosmological mechanisms. Using a blind analysis of data collected over a 813 d live time with DEAP-3600, a 3.3 t single-phase liquid argon-based dark matter experiment at SNOLAB, a search for supermassive dark matter was performed, looking for multiple-scatter signals. No candidate signal events were observed, leading to the first direct detection constraints on Planck-scale mass dark matter. Leading limits constrain dark matter masses between $8.3times10^{6}$ and $1.2times10^{19} text{GeV}/c^2$, and cross sections for scattering on $^{40}$Ar between $1.0times10^{-23}$ and $2.4times10^{-18} text{cm}^2$. These are used to constrain two composite dark matter models.