IAI1123 Captain; shortly after takeoff into icing conditions; experienced landing gear issues and a pitot heat failure. An immediate climb was initiated out of icing conditions but exceeding initial altitude assignment without advising ATC. The gear and pitot heat issues are resolved.

Narrative

During climb out we were given heading of 360 and an altitude of 4;000 FT MSL. Upon selecting gear up; we experienced a malfunction with the gear position resulting in a pitot heat failure. We were in icing conditions and with the chance of our pitot tube icing up; I initiated a climb rate that would get us out of icing conditions as soon as possible. This resulted in an altitude deviation of 1;400 FT which we corrected immediately upon selecting alternate heat for the pitot tubes and began to trouble shoot the gear problem; which we eventually fixed. ATC did ask us our altitude as we did not have time to call them about our situation. We told them that we were leveling at 4;000 FT at which time he said that he showed us at 5;400 FT and gave us a further climb. There were no traffic issues that we were aware of and ATC did not advise us of any. I understand that anytime I have to deviate from an ATC assignment; I should advise them ASAP but under the circumstances time did not allow for a call until we were in safe flight conditions.