The Vela and Geminga pulsars are rotation powered neutron stars, which have been identified in various spectral domains, from the near-infrared to hard $\gamma$-rays. In the near-infrared they exhibit tentative emission excesses, as compared to the optical range. To check whether these features are real, we analysed archival mid-infrared broadband images obtained with the Spitzer Space Telescope in the 3.6--160 $\mu$m range and compared them with the data in other spectral domains. In the 3.6 and 5.8 $\mu$m bands we detected at $\sim$ (4--5)$\sigma$ significance level a point-like object, that is likely to be the counterpart of the Vela pulsar. Its position coincides with the pulsar at < 0.4 arcsec 1$\sigma$-accuracy level. Combining the measured fluxes with the available multiwavelength spectrum of the pulsar shows a steep flux increase towards the infrared, confirming the reality of the near-infrared excess reported early, and, hence, the reality of the suggested mid-infrared pulsar identification. Geminga is also identified, but only at a marginal 2$\sigma$ detection level in one 3.6 $\mu$m band. This needs a farther confirmation by deeper observations, while the estimated flux is also compatible with the near-infrared Geminga excess. The detection of the infrared excess is in contrast to the Crab pulsar, where it is absent, but is similar to the two magnetars, 4U 0142+61 and 1E 2259+586, showing similar features. We discuss X-ray irradiated fall-back discs around the pulsars, unresolved pulsar nebula structures, and pulsar magnetospheres as possible origins of the excesses. We note also possible infrared signatures of an extended tail behind Geminga and of the Vela plerion radio lobes.Comment: 16 pages, 9 figures, 2 table
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