We report measurements of the temperature dependent components of the magnetic penetration depth {lambda}(T) in single crystal samples of YBa_2Cu_4O_8 using a radio frequency tunnel diode oscillator technique. We observe a downturn in {lambda}(T) at low temperatures for currents flowing along the b and c axes but not along the a axis. The downturn in {lambda}_b is suppressed by a small dc field of ~0.25 T. This and the zero field anisotropy of {lambda}(T) likely result from proximity induced superconducting on the CuO chains, however we also discuss the possibility that a significant part of the anisotropy might originate from the CuO2 planes.
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