I will assume that your conditional mean model is well specified. However, weird patterns in the conditional variance equation might very well be due to a poor conditional mean specification. I suggest to make sure you have an adequate model for the conditional mean before going for any exotic conditional variance models.
A GARCH process with coefficients summing up to more than one is not completely implaussible, but certainly it is not very appealing. So what are the alternatives?
is my series a series than simply can't be modelled by GARCH <...> ?
Coefficients of GARCH(1,1) summing up to more than one is an indication that a stationary GARCH$(r,s)$ model is unlikely to fit the data well. However, there is a great variety of GARCH model versions, so you do not have to lose hope. See, for example, this answer and the link to the survey paper there.
If the conditional variance seems to be increasing deterministically (e.g. with a linear trend), you could include a deterministic trend as an exogenous variable in the GARCH model.
So, should I use some other variant of GARCH (IGARCH?) <...> ?
If the GARCH(1,1) coefficients summed up to exactly one, using an IGARCH model would be a natural choice. However, your parameters sum to more than one; hence, IGARCH is not that natural (although more plausible than a stationary GARCH).