  Despite the fact that 1111-type iron arsenides hold the record transition
temperature of iron-based superconductors, their electronic structures have not
been studied much because of the lack of high-quality single crystals. In this
study, we completely determine the Fermi surface in the antiferromagnetic state
of CaFeAsF, a 1111 iron-arsenide parent compound, by performing quantum
oscillation measurements and band-structure calculations. The determined Fermi
surface consists of a symmetry-related pair of Dirac electron cylinders and a
normal hole cylinder. From analyses of quantum-oscillation phases, we
demonstrate that the electron cylinders carry a nontrivial Berry phase $\pi$.
The carrier density is of the order of 10$^{-3}$ per Fe. This unusual metallic
state with the extremely small carrier density is a consequence of the
previously discussed topological feature of the band structure which prevents
the antiferromagnetic gap from being a full gap. We also report a nearly
linear-in-$B$ magnetoresistance and an anomalous resistivity increase above
about 30 T for $B \parallel c$, the latter of which is likely related to the
quantum limit of the electron orbit. Intriguingly, the electrical resistivity
exhibits a nonmetallic temperature dependence in the paramagnetic tetragonal
phase ($T >$ 118 K), which may suggest an incoherent state. Our study provides
a detailed knowledge of the Fermi surface in the antiferromagnetic state of
1111 parent compounds and moreover opens up a new possibility to explore
Dirac-fermion physics in those compounds.
