We report on the results of a multi‐wavelength study of the compact object candidate 1RXS J141256.0+792204 (Calvera). Calvera was observed in the X‐rays with XMM–Newton/EPIC twice for a total exposure time of ∼50 ks. The source spectrum is thermal and well reproduced by a two‐component model composed of either two (absorbed) hydrogen atmosphere models or two blackbodies with temperatures kT1∼ 55/150 eV, kT2∼ 80/250 eV, respectively (as measured at infinity). Evidence was found for an absorption feature at ∼0.65 keV while no power‐law high‐energy tail is statistically required. Using pn and MOS data we discovered pulsations in the X‐ray emission at a period P = 59.2 ms. The detection is highly significant (≳11σ), and unambiguously confirms the neutron star nature of Calvera. The pulse profile is nearly sinusoidal, with a pulsed fraction of ∼18 per cent. We looked for the timing signature of Calvera in the Fermi Large Area Telescope (LAT) data base and found a significant (∼5σ) pulsed signal at a period coincident with the X‐ray value. The gamma‐ray timing analysis yielded a tight upper limit on the period derivative, s s−1 ( erg s−1, B < 5 × 1010 G for magneto‐dipolar spin‐down). Radio searches at 1.36 GHz with the 100‐m Effelsberg radio telescope yielded negative results, with a deep upper limit on the pulsed flux of 0.05 mJy. Diffuse, soft (<1 keV) X‐ray emission about 13 arcmin west of the Calvera position is present both in our pointed observations and in archive ROSAT all‐sky survey images, but is unlikely associated with the X‐ray pulsar. Its spectrum is compatible with an old supernova remnant (SNR); no evidence for diffuse emission in the radio and optical bands was found. The most likely interpretations are that Calvera is either a central compact object escaped from a SNR or a mildly recycled pulsar; in both cases the source would be the first ever member of the class detected at gamma‐ray energies.