  Quantum oscillations in resistivity and magnetization at high magnetic fields
are a macroscopic fingerprint of the energy quantization due to the cyclotron
motion of quasiparticles. In a thin Weyl semimetal, a unique thickness
dependent Weyl-orbit quantum oscillation was proposed to exist, originating
from a nonlocal cyclotron orbit via the electron tunneling between the top and
bottom Fermi-arc surface states. Here, untwinned and high crystalline Weyl
metal SrRuO$_3$ thin films with different thicknesses were grown on miscut
SrTiO$_3$ (001) substrates. Magneto-transport measurements were carried out in
magnetic fields up to 35 T, and quantum oscillations with different frequencies
were observed and compared to the calculated band structure. In particular, we
discovered a frequency $F \approx$ 30 T at low temperatures and above 3 T that
corresponds to a small Fermi pocket with a light effective mass. Its
oscillation amplitude appears to be at maximum for film thicknesses in a range
of 10 to 20 nm, and the phase of the oscillation exhibits a systematic change
with the film thickness. After isolating the well separated frequencies, the
constructed Landau fan diagram shows an unusual concave downward curvature in
the 1/$\mu_0H_n$-$n$ curve, where $n$ is the Landau level index. Based on the
rigorous analysis of the thickness and field-orientation dependence of the
quantum oscillations, the oscillation with $F \approx$ 30 T is attributed to be
of surface origin, which is related to the Fermi-arc surface state originating
from non-overlapping Weyl nodes projected on the film's surface plane. Those
findings can be understood within the framework of the Weyl-orbit quantum
oscillation effect with non-adiabatic corrections.
