Can I build 11' x 5.5' loft with 2x2x8 beam and 2x4 joist?
You will likely experience some bounce using a 2x4 joist even when on 12" centers. I would suggest 2x6 on 12" centers for a nice solid platform floor.

With only a 5'6" width of the platform I think you will have tight work area up there considering a desk on the wall. If the desk is on the long wall I say you have a non-starter situation. If the desk is on the short wall then maybe it is usable but do add a safety railing to the open edge so you do not accidentally fall off. A six foot drop could be downright painful.