An important development in the field of the fractional quantum Hall effect was the proposal that the 5/2 state observed in the Landau level with orbital index n = 1 of two-dimensional electrons in a GaAs quantum well1 originates from a chiral p-wave paired state of composite fermions that are topological bound states of electrons and quantized vortices. The excitations of this state, which is theoretically described by a 'Pfaffian' wavefunction2 or its hole partner called the anti-Pfaffian3,4, are neither fermions nor bosons but Majorana quasiparticles obeying non-Abelian braid statistics5. This has inspired ideas for fault-tolerant topological quantum computation6 and has also instigated a search for other states with exotic quasiparticles. Here we report experiments on monolayer graphene that show clear evidence for unexpected even denominator fractional quantum Hall physics in the n = 3 Landau level. We numerically investigate the known candidate states for the even denominator fractional quantum Hall effect, including the Pfaffian, the particle-hole symmetric Pfaffian and the 221-parton states, and conclude that, among these, the 221-parton appears a potentially suitable candidate to describe the experimentally observed state. Like the Pfaffian, this state is believed to harbour quasi-particles with non-Abelian braid statistics7.