Is it possible to use Workflows to update a field in an invoice which was paid for using PayPal?

Yes, you can.

In the Filter the Triggers section, select the trigger as ‘Edited’ and choose the criteria as ‘Payment Gateway’ is ‘PayPal’ and ‘Status’ is ‘Paid’.

Henceforth, when PayPal is associated to the invoice, and when the invoice is paid, this workflow will be executed.

P.S: But, here is the catch! When an invoice has two payment gateways, say, PayPal and Authorize.net associated with it and if it is paid, the workflow will be executed even if it was paid for using Authorize.net. Also, when a payment is recorded manually for an invoice which is associated with PayPal, the workflow will be executed.

Please ensure that not more than one payment gateway is associated and the invoice is paid online for the workflow to be executed correctly.