You're on a slippery-slope of knock-on upgrades and replacements, and the main limiting factor is your budget.
Yes - your current rear wheel has either a 9 speed hub, or maybe a 10 or 11 speed hub with spacer rings. Take the cassette off and look for one or more metal spacer rings about 50mm across and anything up to 5 mm combined thickness.
There's a chance the freehub body can be swapped to a wider one, but this assumes spares for your hub are available at reasonable prices. Its not unusual to see a replacement freehub body cost as much as the whole hub WITH a freehub.
To update to 10 or 11 you also need at least
- new shifter (hope its not combined with your brake lever)
- new cassette
- new chain (because you don't fit an old chain on a new cassette, and the new chain will be thinner)
You probably also want
- new gear inner and outer, because its coming off anyway so not a lot more work to fit new ones.
At worst case you might need
- replacement rear derailleur (but this would be pretty unlucky if your current one doesn't have enough range once the limit screws are tweaked)
Remember there's absolutely nothing wrong with 9 speed rear. The chain is a little thicker and therefore more long-lasting than 10 or 11 speed.
If you got parts for free from a dead donor bike, it would be worth doing, but otherwise it gets quite expensive for minimal gain.