In this answer: some rather general(izable) advice

_{Pro/con reading it:
(+) applicable to many situations
(–) not based on loads of professional experience/ expert knowledge
(–) may be obvious
(+) but remembering this may still be useful
(+) may be good/ useful advice}

Go with your gut.

Well, I do have a bit more input than that, which might benefit you by triggering some gut feelings – encouraging or discouraging.

(Depending on the (culture and) character of your client and the relationship between the two of you,) I would probably apply this general principle:

When in doubt, try it out. Ask.

Extra carefully, politely!

Chances are, you have more to gain than to loose:

• Even if the chance that they say Yes is small, that outcome might be so useful for you that the opportunity as a whole has weight.

• A 'catastrophic' outcome seems unlikely. They will probably not ditch you just for a silly insolent question (if it even is one (to them)). Even less, since they already showed their high opinion of your work.

• (Even in the unlikely worst case, you will have gained an experience.)

You could also consider

going about it 'cleverly':

Instead of bluntly asking if they'll finance you, you could sort of ask them for advice, telling them that you wonder whether it would be appropriate for a freelancer to ask a client whether they would pay them to learn something like iOS development. Maybe beforehand, sort of casually, mention how interested you would be in learning that.

Advantages of this cautious approach:

• You signal, that you're not confident in this question, so it should seem less grave, in case they might be upset by the request in the first place.

• Your request is clear enough for them to agree if that's what they want.

• If they feel that would be a stupid question to ask, they can decline by phrasing their answer in the form of advice, that this question would not be a good idea. That way they could decline without outright rejecting you and you both 'save face' – it might be less awkward.

It may still come off as unprofessional, but may still be worth it.

Depending on the person's (culture and) character and your relationship, they may prefer a more direct approach. Or none at all.

So all of these are just suggestions to give you ideas – ultimately I'd expect your intuition/ gut to be your best guide on whether you should implement them or what else is the best way for you in this situation to go about it.

Good luck!

PS: If you decide to ask, please share with us  how they reacted! :)