This　paper　focuses　on　the　Rayleigh–Taylor　instability　in　the　system　of　equations　of　the　two-dimensional　nonhomogeneous　incompressible　elasticity　fluid　in　a　horizontal　periodic　domain　with　infinite　height.　First,　we　use　variational　method　to　construct　(linear)　unstable　solutions　for　the　linearized　elastic　Rayleigh–Taylor　problem.　Then,　motivated　by　the　Grenier＇s　idea　in　[10],　we　further　construct　approximate　solutions　with　higher-order　growing　modes　to　the　elastic　Rayleigh–Taylor　problem　due　to　the　absence　of　viscosity　in　the　system,　and　derive　the　error　estimates　between　both　the　approximate　solutions　and　nonlinear　solutions　of　the　elastic　Rayleigh–Taylor　problem.　Finally,　we　prove　the　existence　of　escape　points　based　on　the　bootstrap　instability　method　of　Hwang–Guo　in　[25],　and　thus　obtain　the　nonlinear　Rayleigh–Taylor　instability　result,　which　presents　that　the　Rayleigh–Taylor　instability　can　occur　in　elasticity　fluids　with　small　elasticity　coefficient.　©　2022　Elsevier　Inc.