Current spikes on rising and falling edge of NPN BJT switch
Unless you over dampen the circuit with 100pF capacitors across R6, fast rise and fall times will always cause some type of overshoot and undershoot. If it is bad enough it becomes a decaying ring wave. D1 may play some part in the negative current, as it does bypass the cb junction of Q1. But Schottky diodes do help with overall rise and fall times by lowering the effect of base capacitance. You do not want to over-dampen circuits that switch power, as this can increase overall power consumption itself.

If you look at the output waveform of fast logic IC's or PWM supplies you will notice at least some amount of overshoot and undershoot. The best you can do is make it small enough not to cause ringing or exceed the voltage rating of the driver (Q1 in this case) or cause the next stage (logic gate, etc) to behave in the wrong way.

Run the same test without D1 and see the results. Run the same test with D1 and a 22pF capacitor across the ce junction of Q1. You may find that no compromise is perfect, as having one issue reduced brings up another issue.

NOTE: If you can slow down the rise and fall times of your signal source then overshoot and undershoot problems should go away, but your design using a Schottky diode implies you are doing this intentionally?