There are two likely problems here – one, the texture may be loading before the model, in which case mesh will not be defined when you try to assign it a texture. Two, the model probably contains more than just a single mesh, in which case the root object may not be a mesh itself. Try:
var texture = textureLoader.load('./img/AnimatedArrow.png', function ( map ) {
map.encoding = THREE.sRGBEncoding;
map.flipY = false;
} );
gltfLoader.load( 'models/animated_arrow.glb', function ( gltf ) {
var model = gltf.scene;
model.scale.set(1,1,1);
// ... set position, rotation, scale...
model.traverse( function ( object ) {
if ( object.isMesh ) object.material.map = texture;
} );
scene.add( model );
});