Is flexiblity intrinsic or extrinsic property?

Maybe it is and maybe it isn't. We think of rubber as flexible, but that is only because it contains solvents to make it flexible. When the solvent goes away you have hard rubber, and it can be very hard. We have other types of rubber that don't do that. For instance latex rubber is always flexible without solvents. But a latex balloon is not flexible, at least not much. The air inside stiffens it. So it sometimes depends on what you are talking about and which way you look at it.

The rubber with solvent is intrinsically flexible, but the presence of solvent is an extrinsic factor.

Latex rubber is intrinsically flexible, but the configuration of a balloon and the air inside are extrinsic factors.

"In-" means inside, and "ex-" means outside.