  Graphene-based Josephson junctions provide a novel platform for studying the
proximity effect due to graphene's unique electronic spectrum and the
possibility to tune junction properties by gate voltage. Here we describe
graphene junctions with a mean free path of several micrometres, low contact
resistance and large supercurrents. Such devices exhibit pronounced
Fabry-P\'erot oscillations not only in the normal-state resistance but also in
the critical current. The proximity effect is mostly suppressed in magnetic
fields below 10mT, showing the conventional Fraunhofer pattern. Unexpectedly,
some proximity survives even in fields higher than 1 T. Superconducting states
randomly appear and disappear as a function of field and carrier concentration,
and each of them exhibits a supercurrent carrying capacity close to the
universal quantum limit. We attribute the high-field Josephson effect to
mesoscopic Andreev states that persist near graphene edges. Our work reveals
new proximity regimes that can be controlled by quantum confinement and
cyclotron motion.
