Assuming that the problem occurs in your residuals (as the distribution of the outcome variable itself is usually not a problem), I would be looking to investigate the cause of the problem rather than trying to "fix" it via a transformation or application of a nonparametric model.
If it is the case that there seems to be a trend (e.g., progressively getting more or less normal), or, a clear break between when it goes from normal to not normal, then it suggests a "regime change" of some kind in your data (i.e., the data generating mechanism is changing over time) or some type of missing variable problem.
If it is the case that there is no obvious pattern (e.g., time periods 1 and 3 look normal and time periods 2 and 4 do not) I'd be looking very carefully for a data integrity problem.
A simple way of checking to see if you do have a regime change is to estimate model using only the "normal" time periods and then re-estimate using the other time periods and see what difference occurs. A more complicated approach is to use a latent class model, perhaps with time as a concomitant variable.
As regards your question about nonparametric mixed effects models it kind of depends upon what you mean by nonparametric. If you mean models that do not assume a numeric dependent variable then there are lots of such models (e.g., LIMDEP has quite a few). Also, keep in mind that the violation of the normality assumption will probably only be problematic from an inference perspective if your sample size is small. One way of investigating this would be to try the various transformations discussed in other comments and answers and see if it makes much of an impact on your conclusions.