Atomic beams are a longstanding technology for atom-based sensors and clocks with widespread use in commercial frequency standards. Here, we report the demonstration a chip-scale microwave atomic beam clock using coherent population trapping (CPT) interrogation in a passively pumped atomic beam device. The beam device consists of a hermetically sealed vacuum cell fabricated from an anodically bonded stack of glass and Si wafers. Atomic beams are created using a lithographically defined microcapillary array connected to a Rb reservoir1 and propagate in a 15 mm long drift cavity. We present a detailed characterization of the atomic beam performance (total Rb flux ≈ 7.7 × 1011 s-1 at 363 K device temperature) and of the vacuum environment in the device (pressure < 1 Pa), which is sustained using getter materials which pump residual gases and Rb vapor. A chip-scale beam clock is realized using Ramsey CPT spectroscopy of the 87Rb ground state hyperfine transition over a 10 mm Ramsey distance in the atomic beam device. The prototype atomic beam clock demonstrates a fractional frequency stability of ≈ 1.2 × 10-9/√τ for integration times τ from 1 s to 250 s, limited by detection noise. Optimized atomic beam clocks based on this approach may exceed the long-term stability of existing chip-scale clocks, and leading long-term systematics are predicted to limit the ultimate fractional frequency stability below 10-12.