  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.
