The present disclosure relates to a composition for dyeing or lightening human keratin fibers, comprising: a cosmetically acceptable medium; at least 25% by weight of at least one fatty substance; from 0.5% to 15% by weight of at least one silicate chosen from smectite, vermiculite, stevensite and chlorite clays; at least one dye chosen from oxidation dyes and direct dyes; at least one basifying agent; and at least one oxidizing agent. The disclosure also relates to a dyeing or lightening process using it. Another aspect of the disclosure is multi-compartment devices or kits for obtaining, after mixing together the compositions of the compartments, just before its application, a composition according to the disclosure.