Can I take part of a payment now and the rest later?

Yes. Record any amount against an invoice; the balance stays open for the remainder and the invoice reports as partly paid. Deposits and staged payments do not need a separate document. Part payment is something you record — a customer paying online pays the full outstanding balance, because the share page has no field for a smaller amount.

A part-paid invoice reports as “Partially Paid” even after its due date has passed — the partly-paid state takes precedence over overdue, so use the due date rather than the status if you are looking for late part-paid work.