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We present a search for axion-like polarization oscillations in the cosmic microwave background (CMB) with observations from the Keck Array. A local axion field induces an all-sky, temporally sinusoidal rotation of CMB polarization. A CMB polarimeter can thus function as a direct-detection experiment for axion-like dark matter. We develop techniques to extract an oscillation signal. Many elements of the method are generic to CMB polarimetry experiments and can be adapted for other datasets. As a first demonstration, we process data from the 2012 observing season to set upper limits on the axion-photon coupling constant in the mass range $10^{-21}$-$10^{-18}~\mathrm{eV}$, which corresponds to oscillation periods on the order of hours to months. We find no statistically significant deviations from the background model. For periods larger than $24~\mathrm{hr}$ (mass $m < 4.8 \times 10^{-20}~\mathrm{eV}$), the median 95%-confidence upper limit is equivalent to a rotation amplitude of $0.68^\circ$, which constrains the axion-photon coupling constant to $g_{ϕγ} < \left ( 1.1 \times 10^{-11}~\mathrm{GeV}^{-1} \right ) m/\left (10^{-21}~\mathrm{eV} \right )$, if axion-like particles constitute all of the dark matter. The constraints can be improved substantially with data already collected by the BICEP series of experiments. Current and future CMB polarimetry experiments are expected to achieve sufficient sensitivity to rule out unexplored regions of the axion parameter space.