Hair dyeing process comprising: the application to the hair of a composition comprising at least one pigment and a maleic anhydride ethylenic polymer and of an amine compound chosen from polyamine compounds bearing several primary amine and/or secondary amine groups and amino alkoxysilanes, the ethylenic polymer being derived from the polymerization of: (a) 45% to 95% by weight, relative to the total weight of monomers, of an ethylenic monomer bearing an at least C 8 linear or branched alkyl group; (b) 5% to 25% by weight of maleic anhydride; (c) 0 to 50% by weight of additional monomer chosen from: (i) polydimethylsiloxane silicone monomers bearing a mono(meth)acryloyloxy end group of formula below: in which: - R 8 denotes a hydrogen atom or a methyl group; - R 9 denotes a linear or branched divalent hydrocarbon-based group containing from 1 to 10 carbon atoms and optionally containing one or two ether bonds -O-; - R 10 denotes a linear or branched alkyl group containing from 1 to 10 carbon atoms and especially from 2 to 8 carbon atoms; - n denotes an integer ranging from 1 to 300; (ii) linear or branched C 1 -C 6 alkyl (meth)acrylate or C 6 -C 12 cycloalkyl (meth)acrylate non-silicone monomers; the compositions used containing less than 2% by weight of water, relative to the total weight of the composition, when the amine compound is an amino alkoxysilane. the application to the hair of a composition comprising at least one pigment and a maleic anhydride ethylenic polymer and of an amine compound chosen from polyamine compounds bearing several primary amine and/or secondary amine groups and amino alkoxysilanes, the ethylenic polymer being derived from the polymerization of: (a) 45% to 95% by weight, relative to the total weight of monomers, of an ethylenic monomer bearing an at least C 8 linear or branched alkyl group; (b) 5% to 25% by weight of maleic anhydride; (c) 0 to 50% by weight of additional monomer chosen from: (i) polydimethylsiloxane silicone monomers bearing a mono(meth)acryloyloxy end group of formula below: in which: - R 8 denotes a hydrogen atom or a methyl group; - R 9 denotes a linear or branched divalent hydrocarbon-based group containing from 1 to 10 carbon atoms and optionally containing one or two ether bonds -O-; - R 10 denotes a linear or branched alkyl group containing from 1 to 10 carbon atoms and especially from 2 to 8 carbon atoms; - n denotes an integer ranging from 1 to 300; (ii) linear or branched C 1 -C 6 alkyl (meth)acrylate or C 6 -C 12 cycloalkyl (meth)acrylate non-silicone monomers; (a) 45% to 95% by weight, relative to the total weight of monomers, of an ethylenic monomer bearing an at least C 8 linear or branched alkyl group; (b) 5% to 25% by weight of maleic anhydride; (c) 0 to 50% by weight of additional monomer chosen from: (i) polydimethylsiloxane silicone monomers bearing a mono(meth)acryloyloxy end group of formula below: in which: - R 8 denotes a hydrogen atom or a methyl group; - R 9 denotes a linear or branched divalent hydrocarbon-based group containing from 1 to 10 carbon atoms and optionally containing one or two ether bonds -O-; - R 10 denotes a linear or branched alkyl group containing from 1 to 10 carbon atoms and especially from 2 to 8 carbon atoms; - n denotes an integer ranging from 1 to 300; (ii) linear or branched C 1 -C 6 alkyl (meth)acrylate or C 6 -C 12 cycloalkyl (meth)acrylate non-silicone monomers; (i) polydimethylsiloxane silicone monomers bearing a mono(meth)acryloyloxy end group of formula below: in which: - R 8 denotes a hydrogen atom or a methyl group; - R 9 denotes a linear or branched divalent hydrocarbon-based group containing from 1 to 10 carbon atoms and optionally containing one or two ether bonds -O-; - R 10 denotes a linear or branched alkyl group containing from 1 to 10 carbon atoms and especially from 2 to 8 carbon atoms; - n denotes an integer ranging from 1 to 300; - R 8 denotes a hydrogen atom or a methyl group; - R 9 denotes a linear or branched divalent hydrocarbon-based group containing from 1 to 10 carbon atoms and optionally containing one or two ether bonds -O-; - R 10 denotes a linear or branched alkyl group containing from 1 to 10 carbon atoms and especially from 2 to 8 carbon atoms; - n denotes an integer ranging from 1 to 300; (ii) linear or branched C 1 -C 6 alkyl (meth)acrylate or C 6 -C 12 cycloalkyl (meth)acrylate non-silicone monomers; the compositions used containing less than 2% by weight of water, relative to the total weight of the composition, when the amine compound is an amino alkoxysilane.