I researched about it by searching on Google and reading some bacteriological articles, but I did not get any answer. I also asked some of my teachers, and they were also a bit confused. Some said that it is possible and others said not. So the question remains:
Can a bacterium infect another bacterium? If so, how?
Bdellovibrio bacteriovorus (BV) “infects” other bacteria:
Similar to a virus, BV attacks bacteria such as Escherichia coli (E. coli) by attaching to and entering its prey, growing and replicating within the cell, and then suddenly bursting out, releasing its progeny into the surrounding environment. — How bacteria hunt other bacteria
Note that a lot of the answer to this question hinges on what you mean by "infect". Bdellovibrios (as Laurel has already established that this does happen) do something that does look a lot like infecting eukaryotic cells- but it is still spoken of as "hunting". Why? I am not sure.
There are lots of different ways that bacteria can have negative interactions with each other- though they often resemble "hunting" or "warfare" more than "infection".
Getting a bit more into the "how" of the question, a common mechanism for inter-bacterial negative interaction is the Type VI secretion system. This apparatus is often spoken of as a "weapon" that bacteria can use to kill each other, by adhering to the other cell and shooting toxins in through its membrane. They can use this to great ecological effect.
The type VI secretion system is also used in some contexts for infecting non-bacterial hosts, so by that logic you could argue that it resembles infection.
Update: for a short discussion of inter-bacterial killing (including Bdellovibrio + secretion systems) in the context of biotechnological applications, see here.
