A pilot reported while climbing out as instructed by ATC; they encountered a NMAC with another aircraft 100 ft. above and overtaking which prevented them from avoiding Class Bravo airspace.

Narrative

After departing from ZZZ1 while following the ATC instructions to depart straight out to 3 miles prior to turning north; we spotted a Bonanza; catch us from behind and to our right. The aircraft appeared to be on a closing vector slightly parallel to us. At that point we were approaching ZZZ airspace and almost ready to turn but we then saw the Bonanza turn towards us and close quickly at our altitude but with apparent motion to pass extremely close to us but behind and above. Not knowing whether or not he saw us; we continued on the same course without deviation or altitude change to avoid him and keep him in sight considering he might vector into us if we made a sudden change with such a quick closure rate. He passed behind us and above us by less than 100 ft. After he passed us; we turned north on course but by then we had entered ZZZ Class B by less than a mile but we exited it as quickly as we could. This was the first time I had been in a near air conflict that I saw in advance. I don't know what I would do differently considering the need to keep the aircraft in sight without creating a situation where I lose sight and make sudden movements that occur at the same time as him spotting us suddenly and maneuver into us.