Don't just go fill it!!
There are several things you need to check out just yet.
The biggest questions that need to be answered are;
1. Is it a wetland? To be a wetland it needs to have wetland vegetation, wetland soils, and wetland water characteristics. Being in a floodplain it probably satisfies the third requirement, but you would need to find out about the other two. If it is a wetland depending on the size and quality, you may still be able to fill it in, or you will have options for mitigation, or you will just be stuck.
2. Does the strip of area serve as a drainage path for property other than yours. (i.e. when it rains does stormwater runoff from properties other than yours flow into or across this strip of floodplain) If so then you can't just fill it without it providing a route for this stormwater to get to its ultimate destination. If you fill it in and the stormwater from your neighbor now has nowhere to go and floods his property, you got problems.
3. What type of floodplain is it? FEMA has different classifications (floodplains, floodways, etc.). What it really is will determine what hoops you have to jump through to fill it in.
4. What does the local jurisdiction require? Down here in florida things are so flat and low areas are so connected that it is really like a big bathtub when it rains. All the low areas fill up at the same time. So every bit of storage volume matters. Because of that many cities and counties have a "compensatory volume" rule. What that means is that for every cubic yard of dirt you put in a floodplain, you need to create a cubic yard of floodplain storage somewhere else. (i.e. if you fill in the expensive corner of your property, you will need to dig a hole in the cheap back corner of your property) I doubt you have that rule in Oklahoma but you should ask.
Those are the main questions that need to be answered. There are others.
I don't mean to discourage you. It shouldn't take much to answer these questions, and then you will know what options you have.