We report on the energy-resolved timing and phase-resolved spectral analysis of X-ray emission from PSR J0659+1414 observed with XMM-Newton and NuSTAR. We find that the new data rule out the previously suggested model of the phase-dependent spectrum as a three-component (2 blackbodies + power-law) continuum, which shows large residuals between $0.3-0.7$ keV. Fitting neutron star atmosphere models or several blackbodies to the spectrum does not provide a better description of the spectrum, and requires spectral model components with unrealistically large emission region sizes. The fits improve significantly if we add a phase-dependent absorption feature with central energy $0.5-0.6$ keV and equivalent width up to $approx 50$ eV. We detected the feature for about half of the pulse cycle. Energy-resolved pulse profiles support the description of the spectrum with a three-component continuum and an absorption component. The absorption feature could be interpreted as an electron cyclotron line originating in the pulsar magnetosphere and broadened by the non-uniformity of the magnetic field along the line of sight. The significant phase-variability in the thermal emission from the entire stellar surface may indicate multi-polar magnetic fields and a non-uniform temperature distribution. The strongly pulsed non-thermal spectral component detected with NuSTAR in the $3-20$ keV range is well fit by a power-law model with a photon index $Gamma=1.5pm0.2$.
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