A first experimental test of tracking relativistic charged particles by `drifting' Cherenkov photons in a water-based optical time-projection chamber (OTPC) has been performed at the Fermilab Test Beam Facility. The prototype OTPC detector consists of a 77~cm long, 28~cm diameter, 40~kg cylindrical water mass instrumented with a combination of commercial $5.1\times5.1$~cm$^2$ micro-channel plate photo-multipliers (MCP-PMT) and $6.7\times6.7$~cm$^2$ mirrors. Five MCP-PMTs are installed in two columns along the OTPC cylinder in a small-angle stereo configuration. A mirror is mounted opposite each MCP-PMT on the far side of the detector cylinder, effectively doubling the photo-detection efficiency and providing a time-resolved image of the Cherenkov light on the opposing wall. Each MCP-PMT is coupled to an anode readout consisting of thirty 50 Ohm microstrips. A 180-channel data acquisition system digitizes the MCP-PMT signals on one end of the microstrips using the PSEC4 waveform sampling-and-digitizing chip operating at a sampling rate of 10.24~Gigasamples-per-second. The single-ended microstrip readout determines the time and position of a photon arrival at the face of the MCP-PMT by recording both the direct signal and the pulse reflected from the unterminated far end of the strip. The detector was installed on the Fermilab MCenter secondary beam-line behind a steel absorber where the primary flux is multi-GeV muons. Approximately 80 Cherenkov photons are detected for a through-going muon track in a total event duration of ~2 ns. By measuring the time-of-arrival and the position of individual photons at the surface of the detector to $\le$100~ps and a few mm, respectively, we have measured a spatial resolution of $\sim$ 15~mm for each MCP-PMT track segment, and, from linear fits over the entire track length of $\sim40$~cm, an angular resolution on the track direction of $\sim60$~mrad.