  We investigate the dynamical friction (DF) acting on circularly-moving
perturbers in fuzzy dark matter (FDM) backgrounds. After condensation, FDM is
described by a single wave function satisfying a Schr\"odinger-Poisson
equation. An equivalent, hydrodynamic formulation can be obtained through the
Madelung transform. Here, we consider both descriptions and restrict our
analysis to linear response theory. We take advantage of the hydrodynamic
formulation to derive a fully analytic solution to the DF in steady-state and
for a finite time perturbation. We compare our prediction to a numerical
implementation of the wave approach that includes a non-vanishing FDM velocity
dispersion $\sigma$. Our solution is valid for both a single and a binary
perturber in circular motion as long as $\sigma$ does not significantly exceed
the orbital speed $v_\text{circ}$. While the short-distance Coulomb divergence
of the (supersonic) gaseous DF is no longer present, DF in the FDM case
exhibits an infrared divergence which stems from the (also) diffusive nature of
the Schr\"odinger equation. Our analysis of the finite time perturbation case
reveals that the density wake diffuses through the FDM medium until it reaches
its outer boundary. Once this transient regime is over, both the radial and
tangential DF oscillate about the steady-state solution with an exponentially
decaying envelope. Steady-state is thus never achieved. We use our results to
revisit the DF decay timescales of the 5 Fornax globular clusters. We also
point out that the inspiral of compact binary may stall because the DF torque
about the binary center-of-mass sometimes flips sign to become a thrust rather
than a drag (abridged).
