I have a power drill that looks like the picture below. I've had it for years, but recently I noticed that it squeaks when I use it (I only really hear the squeaking while it's winding down, when it's spinning full speed it's too loud to hear anything else). My question is: should I try to lubricate it somehow? Is this an indication that I should get a new drill, or just be kinder to my old one?
The drill you display is for DIY use (as opposed to professional use), so it's designed to be durable enough for years of typical DIY usage. So it has no user-serviceable parts - even its brushes will last for about one hundred hours of motor running and you're not expected to think of what happens to them after that one hundred hours.
That's why you really have two options - either just continue using it or bring it to a service center and prepare to pay a fortune for servicing.
One possible reason of noise is that after mild (thus permissible) wear accumulation it just started to sound differently.
If you have it apart, I would place a single drop of machine oil (I use a spindle oil, which I also use in the SO's quilting machine) on the rear bearing. Don't let it get on the brushes.
I have a Craftsman drill I bought in 1969, and still use from time to time. A drop of oil every 5 or 10 years served it well. The gearbox was readily serviceable, and twice I have cleaned out the old grease and installed a premium synthetic chassis grease.
However, a drill such as you have was not designed with servicing in mind, and is intended for limited use. If you use it allot, and you like it, you can show it some love.