First: we don't really say that arguments are true or false. Statements are true or false, but arguments have different kinds of properties.
One of those properties is, as you are obviously aware of, validity. However, another important property is well-foundedness, which means that the premises are true (or, for more practical everyday purposes, plausible or acceptable).
Well-foundedness is important, because if I am allowed to just assume anything as my premise, I can (validly!) argue for anything. For example:
"All dogs are purple. Foofy is a dog. Therefore, Foofy is purple"
This argument is logically valid, but not well-founded. And indeed, as such it is a bad argument.
... which is probably just what you were looking for when you said you wanted a valid but 'false' argument. Indeed, instead of saying that arguments are true or false, you can say they are good or bad (and of course anything in between: pretty good, pretty bad, ho-hum, excellent, terrible, etc.)
A special kind of 'bad' argument is something like this:
"Bananas are yellow. Therefore, bananas are yellow"
Interestingly, this argument is logically valid, and its premises are true (well, not in my local supermarket, which for some reason thinks that I would like to purchase their still green bananas, but you get the point). However, it is what you will recognize as a circular argument ... which is bad. OK, but why exactly is it bad? Well, think about it: why would someone be looking for an argument as to whether bananas are yellow or not? Presumably it is exactly because such a person doesn't know whether bananas are yellow or not. And we really shouldn't be assuming something that, to this person, is not acceptable ... which is another reason why for real life purposes, it may be more useful to define well-foundedness as 'the premises are acceptable' rather than 'the premises are true'.