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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 Schrödinger-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 $σ$. Our solution is valid for both a single and a binary perturber in circular motion as long as $σ$ 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 Schrödinger 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).