If we accept the conclusion that Blade Runner 2049 pushes us towards as correct, then it's clear that K is:
A modern/recent generation replicant who was engineered without the capacity for independent thought, action, and disobedience that made the earlier series of replicants 'problematic', and not the offspring of two earlier-generation replicants who do possess that capacity (and would presumably pass it on).
It follows that at no point should he be capable of deliberate deception such as lying, or even of disobeying a direct order (from an authorized source, presumably).
However he does tell Joshi that he's "dealt with" (I think; don't recall his exact words) the issue that she tasked him with, and she seems quite confident that this can only mean precisely what she thinks it means; although obviously it didn't.
Was K lying when he relayed that bit of information to Joshi, or had her earlier instructions been sufficiently vague that there was a way for K to truthfully claim to have dealt with the issue without actually having done so?
Furthermore, if Joshi hadn't:
Warned K that he had 48 hours to hide and effectively ordered him to run for his life
Would he have done so anyways or would the entire second half of the movie been completely averted?
Does K demonstrate independence on a level that his origin says he's not supposed to possess, or can all of his actions be explained within the context of obeying orders from a recognized superior (albeit sometimes with broad discretion as to how)?