He followed his first-innings 108 with 105 not out, including a six and 11 fours, and shared a stand of 114 with Chesney Hughes (50).
Wayne Madsen weighed in with 43 before he was bowled by Ivan Thomas (2-68).
The hosts eventually declared on 234-5 following a rain delay, and Kent progressed to 30-0 by the close.
Godleman's fine knock was only the fifth time a Derbyshire batsman has hit three centuries in successive innings and the first since Kim Barnett in 1990.
His half-century took 80 balls and he moved carefully to three figures with a single off spinner James Tredwell.
The teams were forced to take an early tea at 15.15 BST because of bad light rain and Derbyshire opted to declare when play resumed two and a half hours later, but they were unable to separate Daniel Bell-Drummond (18 not out) and Rob Key (11) before stumps.
Derbyshire batsman Billy Godleman hit a third successive Championship century as they set Kent a target of 329 on day three at the County Ground.