General aviation pilot reported a near miss with another aircraft while climbing out from a non-towered airport in daylight conditions under IFR control. The pilot maneuvered the aircraft based upon a TCAS traffic alert only and climbed through his assigned altitude; with no visual contact of the other aircraft; and ATC asked why the initial assigned altitude had been exceeded.

Narrative

IFR flight plan filed GAI/ZZZ. Calling rolling Runway 32 GAI with departure clearance (heading and altitude). At 700 ft. AGL started turn onto heading direct EMI per clearance. Half way into turn; passing 1400 ft. AGL received ATA alert on TCAS that an aircraft was in front and below at 150 ft. and climbing into my flight path from my right in blind spot. I immediately increased power and ROC; and flying and looking for traffic visually and on MFD. At full power I estimate my ROC to have been 2500 FPM which put me through my initial assigned altitude. Very quickly before I could call approach/departure at ZZZ; the controller came on frequency to ask my assigned altitude; which I replied 030 after which he inquired why I had exceeded the assigned altitude. I informed him that I had experienced a TA alert and identified a threat on the MFD in close proximity to my flight path and took evasive action. It all happened so fast that I never realized that I had exceeded the initial assigned altitude but on his call stopped the climb at 3300 ft. MSL I indicated. He then cleared me up to 8000 ft. MSL and advised me of a potential PD (Pilot Deviation) and asked me to copy a telephone number when ready. On arrival at destination; ZZZ; I called ZZZ Approach and advised the responder of the circumstances leading to the incident. He asked for my particulars and contact details and said he would review ADSB and radar records and revert to me if follow up was required.