This web page produced by SAS provides an example of fitting an LMM using PROC MCMC in SAS. Additional examples of mixed-effects models produced by SAS can be found here.

Additional interesting comparisons of Bayesian approaches in R and SAS can be found here.

PROC MIXED in SAS does have a PRIOR statement enabling Bayesian analysis for simple variance components models (e.g., the Rat Pup data), but more advanced models will require the use of PROC MCMC. For an example of using the PRIOR statement, see this paper.