When a credit card is used to make a purchase/payment the cardholder has the ability to chargeback that transaction for up to 180 days with the issuing bank. If they have not received the product and/or service as promised they will contact the issuing bank.

You should set up your contracts so there's clarity on payments and milestones per payment. In that way, you and your customer are already agreed on what payments are owed if you become in breach of contract, which is apparently what this client believes you have done. Additionally, if a client then attempts to rescind the payment, you have the signed contract to prove to the bank or credit card company that the chargeback is not appropriate (unless it is). For each milestone, the payment is confirmation that the milestone is complete. You can also request confirmation with a set number of days defined where any bugs or issues may be reported upon delivery or after the milestone date. And state in the contract that once the set number of days transpires without requests for fixes, that this is defined to mean that the client accepts and is satisfied with what you delivered. Some contracts call for x% to be paid upon contract start, and then the remainder to be paid over 1 or more additional milestones. The contract should list clearly the options that the client and you have to sever the relationship and what this means as far as remaining payments.

Chargebacks are for consumers who do not receive the goods they were promised by a vendor. It's typically tied to credit card purchases.

Please feel free to give me a call if you have further requirements that cause you to have questions on the above.

It is more about setting correct expectations and have policies in place to adjust if expectations and milestones are not met.

First one would have to define the charge-back policy within their contracts. Then a determination of fault would have to be identified by either parties in that contract.
The problem with using credit cards for payments is the ability to have the payer request a charge back based on evidence "X". Since you missed a deadline, does it state in your contract agreement or other form of project acceptance that there would be a charge back if the deadline was missed? In this day and age I do believe that every business should have ways to protect themselves, especially if you are dealing with international clients.
Was there any correspondence from you as to the potential to miss a deadline? There are too many unknown variables to determine the proper placement of a policy such as this. Most standard agreements will have a "mediation before litigation" clause in them to avoid the predicament you are currently in. Best of luck.