In short, evolution is perfectly possible provided you crack the incredibly hard problem of non-water DNA. However, it's going to be veerrrrryyy slow.

Darwin's theory of Natural Selection is all about nonrandom genetic changes that mean a branch of a species is better adapted to living in a certain environment. Early AI took a similar approach, so this is definitely not limited to water-based organisms.

So considering that evolution as a process is invariant to biochemistry, let's consider the rate at which genetic mutations would occur within an ammonia-based life form compared to a water-based one. Here's the first problem:

Scientists have just shown water is the key to binding DNA strands. The reason is to do with the polarity of water, which is something that's theoretically possible with ammonia but I wouldn't even know how to begin approaching that problem at scale. I'm aware you said the 'how' of life doesn't matter. If you can form ammonia-based DNA, evolution is (relatively speaking) a cinch with genetic mutations.

With this in mind, the (far) hydrogen bonds between ammonia molecules reduce its power to concentrate non-polar molecules in a hydrophobic way. This means ammonia DNA is going to have to be far more stable, stronger or just much less error-prone in some way to counteract for the fact that it's more likely to fall apart than water-based DNA. The likely effect of this is the evolutionary matrix (described here) can't afford to take as many chances as water-based DNA. Thus, the chance of mutations is going to be significantly diminished to promote chemical stability.

Phew. In short, the weaker hydrogen bonds in ammonia are going to cause you a list of problems, but if you get ammonia-based DNA worked out it will probably produce natural selection-esque evolution at a rate far slower than that on Earth.

EDIT: Just saw your edit on breathing chlorine. Chlorine respiration? Oh man. You're really going to screw up my calculations here.

In short, evolution is perfectly possible provided you crack the incredibly hard problem of non-water DNA. However, it's going to be veerrrrryyy slow.

Darwin's theory of Natural Selection is all about nonrandom genetic changes that mean a branch of a species is better adapted to living in a certain environment. Early AI took a similar approach, so this is definitely not limited to water-based organisms.

So considering that evolution as a process is invariant to biochemistry, let's consider the rate at which genetic mutations would occur within an ammonia-based life form compared to a water-based one. Here's the first problem:

Scientists have just shown water is the key to binding DNA strands. The reason is to do with the polarity of water, which is something that's theoretically possible with ammonia but I wouldn't even know how to begin approaching that problem at scale. I'm aware you said the 'how' of life doesn't matter. If you can form ammonia-based DNA, evolution is (relatively speaking) a cinch with genetic mutations.

With this in mind, the (far) hydrogen bonds between ammonia molecules reduce its power to concentrate non-polar molecules in a hydrophobic way. This means ammonia DNA is going to have to be far more stable, stronger or just much less error-prone in some way to counteract for the fact that it's more likely to fall apart than water-based DNA. The likely effect of this is the evolutionary matrix (described here) can't afford to take as many chances as water-based DNA. Thus, the chance of mutations is going to be significantly diminished to promote chemical stability.

Phew. In short, the weaker hydrogen bonds in ammonia are going to cause you a list of problems, but if you get ammonia-based DNA worked out it will probably produce natural selection-esque evolution at a rate far slower than that on Earth.