We report on specific heat measurements on clean overdoped BaFe2(As1−xPx)2 single crystals performed with a high resolution membrane-based nanocalorimeter. A nonzero residual electronic specific heat coefficient at zero temperature γr = C/T |T →0 is seen for all doping compositions, indicating a considerable fraction of the Fermi surface ungapped or having very deep minima. The remaining superconducting electronic specific heat is analyzed through a two-band s-wave α model in order to investigate the gap structure. Close to optimal doping we detect a single zero-temperature gap of ∆0 ∼ 5.3 meV, corresponding to ∆0/kBTc ∼ 2.2. Increasing the phosphorus concentration x, the main gap reduces till a value of ∆0 ∼ 1.9 meV for x = 0.55 and a second weaker gap becomes evident. From the magnetic field effect on γr, all samples however show similar behavior [γr(H) − γr(H = 0) ∝ H n , with n between 0.6 and 0.7]. This indicates that, despite a considerable redistribution of the gap weights, the total degree of gap anisotropy does not change drastically with doping.