Hello all, I'm quite fond of my 2014 Volt...
As stated above by others, try to negotiate before they find out how many miles you have. The worst they can say is no. Then call back next week. And the next... Many people here have had success doing this.
According to Black Book, a 2014 Volt (base - you didn't state what you have) with 56,000 miles, is worth $19,000 (extra clean) , $17,300 (clean) down to $15,625 (average). Potential add-ons are Rearview Camera System - $275, Navigation - $500, and Leather - $700.
I'll leave you to calculate where your Volt's value is from there but I do believe knowledge is power in negotiating.
The great news is that, as you stated above, you love the car! If you can get a deal, great. If they insist on you paying the contract residual, then you have a choice to make. Either way, you're going to spend at least $5k pretty soon.
Let us know what you decide.