Context. Finding solar siblings, that is, stars that formed in the same cluster as the Sun, will yield information about the conditions at the Sun’s birthplace. Finding possible solar siblings is difficult since they are spread widely throughout the Galaxy. Aims. We search for solar sibling candidates in AMBRE, the very large spectra database of solar vicinity stars. Methods. Since the ages and chemical abundances of solar siblings are very similar to those of the Sun, we carried out a chemistry- and age-based search for solar sibling candidates. We used high-resolution spectra to derive precise stellar parameters and chemical abundances of the stars. We used these spectroscopic parameters together with Gaia DR2 astrometric data to derive stellar isochronal ages. Gaia data were also used to study the kinematics of the sibling candidates. Results. From about 17 000 stars that are characterized within the AMBRE project, we first selected 55 stars whose metallicities are closest to the solar value (−0.1 ≤ [Fe/H] ≤ 0.1 dex). For these stars we derived precise chemical abundances of several iron-peak, α- and neutron-capture elements, based on which we selected 12 solar sibling candidates with average abundances and metallicities between −0.03 and 0.03 dex. Our further selection left us with four candidates with stellar ages that are compatible with the solar age within observational uncertainties. For the two of the hottest candidates, we derived the carbon isotopic ratios, which are compatible with the solar value. HD 186302 is the most precisely characterized and probably the most probable candidate of our four best candidates. Conclusions. Very precise chemical characterization and age estimation is necessary to identify solar siblings. We propose that in addition to typical chemical tagging, the study of isotopic ratios can give further important information about the relation of sibling candidates with the Sun. Ideally, asteroseismic age determinations of the candidates could solve the problem of imprecise isochronal ages.