Is renormalizability only a problem in perturbation theory? As far as I know, renormalization is needed when a scattering amplitude is divergent at some order of the coupling constant in a perturbation theory. So my question is whether the divergence (and thus the problem of renormalizibility) is only a byproduct of the perturbative method, or it still appears in non-perturbative theory?
 A: Renormalisation goes beyond simply removing infinities from a calculation; there is a reason that this problem arises in the first place, and this is the fact that the coupling changes with scale.
In fact, essentially all the parameters of a theory depend in some way on the energy scale of a process being considered, such as the masses as well. The dependence and the way the theory is going to behave with changes in scale is encoded in the renormalisation group flow.
It is not simply valid either to a specific order in perturbation theory. It has not been accomplished with many other theories, but for super-Yang Mills, we have the exact beta function,
$$\beta(\alpha) = -\frac{\alpha^2}{2\pi} \left[3T_G - \sum_i T(R_i) (1-\gamma_i)\right] \left(1-\frac{T_G\alpha}{2\pi} \right)^{-1}$$
depending on some group theoretic parameters, due to Novikov et al. Thus, we can in theory predict the exact change in the coupling constant when going from one energy scale to another and this is to be understood as something physical, not a calculational tool - we would observe that indeed the coupling is different in experiment. 
Now, one may argue that since this issue is over the coupling constant, maybe there is some arbitrariness associated to it, to the perturbative method as you say. However, the coupling has not been introduced artificially as a 'small' parameter to be able to do perturbation theory, the notion of a coupling goes back to classical mechanics, and it is not artificially inserted, but a necessity as far as we know to describe the theory. 
A: No, it isn't. Renormalizability is basically a possibility to define the QFT on continuous spacetime because long-distance and short distance physics are decoupled.
UV divergences which really are UV ambiguities are a general consequence of naive definition of the QFT on the continuous spacetime. If you try for example to write the non-perturbative Hamiltonian in the interacting theory they will arise as powers of $\delta$-functions which are ill-defined objects. Similarly you can try to define the path integral and find that there are ambiguities in the definition of the measure. You can complete their definition in infinitely many ways that corresponds to the freedom of regularization of the continuous spacetimes. Generally speaking this ambiguity doesn't disappear and you have to introduce infinitely many parameters.
For example you can introduce a lattice and quantize some theory on the lattice. However when you consider the continuous limit there are absolutely no guarantees that your model will forget all the details of the lattice (e.g. was it rectangular or hexagonal etc). Moreso, generally speaking it will not forget.
However when you consider RG flows it happens that some of them concentrate in IR near certain finite dimensional subspaces of the theory space. So QFTs that correspond to those finite dimensional subspaces forget all the details about UV except finite number of parameters. Those are renormalizable theories.
