  The MICROSCOPE experiment was designed to test the weak equivalence principle
in space, by comparing the low-frequency dynamics of cylindrical "free-falling"
test masses controlled by electrostatic forces. We use data taken during
technical sessions aimed at estimating the electrostatic stiffness of
MICROSCOPE's sensors to constrain a short-range Yukawa deviation from Newtonian
gravity. We take advantage of the fact that in the limit of small
displacements, the gravitational interaction (both Newtonian and Yukawa-like)
between nested cylinders is linear, and thus simply characterised by a
stiffness. By measuring the total stiffness of the forces acting on a test mass
as it moves, and comparing it with the theoretical electrostatic stiffness
(expected to dominate), it is a priori possible to infer constraints on the
Yukawa potential parameters. However, we find that measurement uncertainties
are dominated by the gold wires used to control the electric charge of the test
masses, though their related stiffness is indeed smaller than the expected
electrostatic stiffness. Moreover, we find a non-zero unaccounted for stiffness
that depends on the instrument's electric configuration, hinting at the
presence of patch-field effects. Added to significant uncertainties on the
electrostatic model, they only allow for poor constraints on the Yukawa
potential. This is not surprising, as MICROSCOPE was not designed for this
measurement, but this analysis is the first step to new experimental searches
for non-Newtonian gravity in space.
