The client is taking advantage of you. Plain and simple.
If "payment is not an issue" as the client states, then the client would have no problem paying you what you are owed to the current point. Whatever that may be. If you had an agreement of 30% at this particular stage... then demand that 30% before doing more.
In my experience, any client that states "payment is not an issue" but then doesn't pay when invoiced... won't pay you if you allow that to happen. At that stage they are merely trying to "milk" you for anything they can get.
Stick to your guns, require payment for more work. There's no point in trying to placate a client with further work if that client is already showing a inclination to take advantage of you. Business is a two way street, the client trusts you to work, you trust the client to pay. You've held up your end. You've shown you are trustworthy and will complete what you agree to complete. Now it's up to the client to prove they can be trusted.... don't do anything more for the client until they show a willingness to hold up their end.
Based on comments on various answers, if there's no contract -- there's nothing you can do to force payment. But also be aware, that an "agreement " is a contract. I mean "an agreement between two or more parties" is the definition of a "contract". If you have written proof (emails) that the client approved pricing, and then approved work.... you have grounds to be firm.
Note that this is a BIG reason I avoid international clients in many instances. It's nearly impossible to collect from someone if they are in another country, let alone on another continent. And that may also be why some clients don't hire locally - they know you can't collect. I'll take on UK and Canadian clients that's it (I'm US based). The UK has some terrific leverage to use against companies that owe more than 5,000£ if the debt can be proven. But, I only discovered that fact about the UK due to a delinquent UK client and researching collection options.
If you do come across international clients, I strongly recommend at least a 50/50 payment schedule. 50% up front, then 50% upon completion (before final file delivery). If possible, 100% up front is best for international clients.