The temperature dependence of the in-plane, lambda_{parallel}, and interplane, lambda_{perp}, London penetration depth was measured in the metal-free all-organic superconductor beta-ET (see title) ($T_c approx$ 5.2 K). lambda_{parallel} ~T^3 up to 0.5 Tc, a power law previously observed only in materials thought to be p-wave superconductors. lambda_{perp} is larger than the sample dimensions down to the lowest temperatures (0.35 K), implying an anisotropy of lambda_{perp}/lambda_{parallel} ~ 400-800.
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