Automatic deletion of specific form data after submission

Hi there,
Bit of an usual question. A client of mine has a will planning form which contains user defined financial information. The form states to the end user that none of this information will be stored by my client. So they've asked whether there's an option to have certain fields automatically deleted after submission. I can't think of how this would be possible for individual fields and even deleting the whole form apart from a admin user going in and deleting the whole entry from within Wordpress database. So was just wondering if there were any options I'm unaware of to tackle this.
Many thanks,
Phil

Thanks David. That's really helpful. I'll take a look at some of the code samples. Just so I understand this correctly, if I use gform_after_submission to strip out specific fields after submission, by definition does this mean these fields will be removed from all confirmation emails and the entries log in the database? My client has since asked if the code could be more specific and only remove particular fields from the admin confirmation email and database entries log but send the entire form with all fields included for the end user confirmation email. Many thanks, Phil

Thanks Chris. That's great!