We consider the Kirchhoff equation for a vibrating body, in any dimension, in the presence of a time-periodic external forcing with period 2π/ω and amplitude ∈. We treat both Dirichlet and space-periodic boundary conditions, and both analytic and Sobolev regularity. We prove the existence, regularity and local uniqueness of time-periodic solutions, using a Nash-Moser iteration scheme. The results hold for parameters (ω, ∈) in a Cantor set with asymptotically full measure as ∈ → 0
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