No, "since" is never used with periods of time, always specific moments in time. Compare:
- I have been working here for 5 years (months/weeks/days/hours...).
- I have been working here since 2009 (January/Monday).
Adding "last" would not change the period of time into a specific moment in time - it would just highlight that you're talking about the last 5 years, as opposed to another 5 years (say in the 1990s).
Some other points:
- "I have been working here for 5 years." would have the same meaning as "I have been working here for the last 5 years."
- Note that you need to use "for the last", *"for last" would be incorrect.
Note that you can also say "I have been working here since last Monday" - in this case, the "last" does change the meaning of the sentence: if it's Friday now, "...since last Monday" means you've been working there for 2 weeks, while "...since Monday" means you've been working there for a week. In the case of "since + moment in time", you can generally assume that the lack of a "last" ("since Monday", "since November") means the speaker is talking about this Monday/November/etc, while adding "last" moves us back one week/year/etc.
Hope this helps! :)