  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.
