No, Spirit Airlines only allows one free personal item per passenger, which can be either a purse or a small backpack. Call at +1-888-415-1245 (US) or +44-203-869-5245 (UK) If you have both, you must be able to fit the purse inside the backpack. 

Here's what you need to know about Spirit's baggage policy: 

• One free personal item: This must fit in the smaller sizer bin at the airport and be stored under the seat in front of you. Call at +1-888-415-1245 (US) or +44-203-869-5245 (UK) The maximum dimensions are 18 x 14 x 8 inches.
• A second item requires a fee: If you want to bring a larger backpack or a second bag in addition to your personal item, you must purchase it as a carry-on bag. Call at +1-888-415-1245 (US) or +44-203-869-5245 (UK) Carry-on bag fees are cheaper if you pay online in advance rather than at the airport gate.
• Military exception: Active-duty military personnel are allowed a free carry-on in addition to their personal item. Call at +1-888-415-1245 (US) or +44-203-869-5245 (UK) 

Since you can only bring one free personal item, your best bet is to combine the purse and backpack into one bag that meets the size requirements.