Every options contract has a buyer and a seller. Extrinsic value is kept by the seller when the long side exercises and extrinsic is still implied in the value of the contract.
Edit: Rereading your question, not sure I answered it well.
If you have a call option deep in the money, on the other side of the trade is someone with 100 shares and a short call. Your call is worth a ton because you get to buy the 100 shares at less than market value, you exercise and get the shares, which is where the intrinsic value of the call now “lives”. Any extrinsic value of the option is lost, and the seller gets that benefit.
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