Functional problem and verifying solutions
It appears that you are asking whether FP = FNP.  That is an open problem.  It is known that FP = FNP if and only if P = NP.  I suspect that most theoreticians expect that it's likely that P != NP, which would imply FP != FNP.  If FP != FNP, there exists a function problem whose solutions can be verified in polynomial time, but where you can't find a solution in polynomial time.  In particularly, any FNP-complete problem -- such as FSAT, say -- will have that property, if FP != FNP.  So, the FNP-complete problems are particularly good candidates for having the property you articulate.  See https://en.wikipedia.org/wiki/Function_problem for an overview.