Yes you can, but it seems you're a bit early. See below another forum members experience circa 2010:
"All in all, based on previous rulings of the court, my conclusion was that you can use mandamus if the delay in your immigration file has been "unreasonable". To be regarded as unreasonable, a delay must meet the following three conditions:
(1) the delay in question was longer than the nature of the process required, prima facie;
(2) the applicant and his or her legal counsel were not responsible for the delay;
(3) the authority responsible for the delay did not provide satisfactory justification.
It seems that a period of more than four or five years has been accepted as "the nature of the process required" in previous rulings. There are even previous cases involving permanent residence applications where delays between three and five years were found to be prima facie longer than required (see cases of Platonov, supra, Kalachnikov, supra and Hanano, supra).
You can go to http://decisions.fct-cf.gc.ca/en/index.html and use the keywords "mandamus" and "immigration" and "CSIS" to find similar cases.
Be careful about the costs. I got quotes ranging from $2000 to $8000. I finally went with a lawyer that had several successful cases and charged me $2000 for a mandamus application. Our agreement was that if the matter was not solved before the court date, I would pay an additional $1000 to him to appear in the court."
