If not, why? Doesn't it make flying safer? It seems to me that automating a small plane should be easier than automating a car (excluding dealing with Air Traffic Control). Is it too expensive?
Many small, light aircraft are equipped with an autopilot today. They can range from very simple autopilots, without altitude selector and so on,
Source, ST SYS 30, $ 12,000
to advanced systems with altitude selector, automatic trim and so on.
Source, Bendix/King KFC 225, $ 15,000 - 42,000
And with this GPS, feeding the autopilot with data, it has the abilities of an airliner.
Source, Garmin® GTN™ 750, $ 17,000
You have to know, the systems in the general aviation industry are getting better and better. Of course they are way below the complexity of the avionics of airliners, but the features are the same if not more.
For example: A captain of an airliner has to review or change the flight plan with the FMC, respectively MCDU on Airbus, a not very straight-forward device to control the plane.
Of course, the FMS has a lot more capabilities than a GA GPS, but one of it's primary features is the input of the flight plan. And in that, from the usability-view, the Garmin® GPS has the better solution: An all-in-one navigation device with touchscreen input.
Yes. Small airplanes can have autopilots. They are very expensive (~$20,000) but are occasionally added to planes as aftermarket modifications or as updates to existing systems.
Autopilots in small planes range from simple wing levelers to fully coupled systems that can use GPS systems fly instrument procedures all the way to published minimums (200' above the ground in some cases).
I owned a small four seat plane with an autopilot that could hold altitude and a heading -or- a VOR/LOC course -or- a GPS track. It was considered a relatively basic system.
Does it make flying safer? That is a very interesting question that is open to debate. The great thing about an autopilot is that when turned on, it frees up the part of your mind you are using to control the aircraft to actually think about other things such as where you are going. The bad thing about autopilots is that they are complex and need to be set correctly (and can fail). So you need to be more familiar (or current in pilot speak) with the make and model of autopilot, GPS, whatever, in your aircraft. Years ago when I learned to fly the CAA here in the UK didn't encourage the use of Autopilots or GPS. Should they? Like I said it cuts both ways. Before I get loads of comments saying what a dinosaur I am etc etc. I would like to refer to one example I read about in a light aircraft where the autopilot was set to altitude hold in cloud and the engine failed. The autopilot held the altitude and the aircraft stalled, spun, and crashed. For sure they have their place but I just don't get the button I can press and then I don't need to do anything mentality. Maybe thats just me.
To answer the why part of your question, turn it around. Most of us who fly small airplanes do so at least in part because we enjoy doing the flying. Why on Earth would we want an expensive device to keep us from doing what we are paying good money to do? We might as well just fly commercial :-(
Indeed, I honestly don't see why Tesla is wasting all that money trying to develop self-driving cars. It just ensures that I'll never buy one.