We report upper critical-field measurements in the metal-free-all-organic superconductor β″-(ET)2SF5CH2CF2SO3 obtained from measuring the in-plane penetration depth using the tunnel diode oscillator technique. For magnetic field applied parallel to the conducting planes the low-temperature upper critical fields are found to exceed the Pauli limiting field calculated by using a semiempirical method. Furthermore, we found a signature that could be the phase transition between the superconducting vortex state and the Fulde-Ferrell-Larkin-Ovchinnikov state in the form of a kink just below the upper critical field and only at temperatures below 1.23 K.