It is still an open question as to whether or not the theory that is commonly used for calculations in high energy physics can be interpreted as an actual field theory in the sense of being able to produce well defined predictions for experiments at arbitrarily high energy-momentum scales – or equivalently arbitrarily short distance-time scales (which is why such theories are sometimes called strictly local field theories).
In fact no interacting strictly local quantum field theory has yet been constructed in a four dimensional spacetime, and some theories which do make sense in two and three dimensions have been shown not to do so in four. But the Yang-Mills type theory that underlies the Standard Model has not been either excluded or shown to be incompatible with the conditions of such a theory.
Current approaches to actual calculations combine perturbation theory for versions of the theory with cutoffs with a non-perturbative “renormalization group” analysis of how the predictions vary with changes of cutoff and coupling parameters. These may be sufficient for most practical purposes but answers which portray that as somehow winning a competition against the question of what mathematical framework is appropriate for working at all scales are in my opinion both misleading and pathetically misguided.