What a ridiculous question! Of course not!... Oh, wait... Errr.

Yes

There are more than one fish that can climb trees. [Reference: Quantum Biologist, which is only a blog, so more evidence below.]

One example is the Mangrove Rivulus (proper name coming), which [description coming]. There's even a YouTube video so you can see for yourself.

[Reference coming]

What a ridiculous question! Of course not!... Oh, wait... Errr.

Yes

There are more than one fish that can climb trees. [Reference: Quantum Biologist, which is "only" a blog, I dug further.]

One example that the Quantum Biologist describes is the Mangrove Rivulus.

They lose their aggressive habits, climb the nearest tree or rotting log, and stuff themselves into the cracks and insect-bored holes. Their physiology changes: instead of breathing through their gills, they breathe (and excrete waste) through their skin. They can stay that way, a literal fish out of water, for over two months, just waiting in the trees for rain to come.

Source: Quantum Biologist

I found several sources (including Reuters) that confirmed it could live outside of the water, inside damp logs and crab burrows for long periods during dry spells, but I couldn't find any that use the word "climb". The Florida Museum of Natural History use the term "slithering and flipping" in its search for a place to hide. 

There's a YouTube video showing the fish inside a log, but the quality is very poor, and hardly proof of anything.

The US Department of Commerce has declared it a Species of Concern.