Light reflecting off rain from bike light

Yes.

LED lights are often fed from a circuit that causes them to flash so quickly they look continuous.

What you are seeing is the stroboscopic effect.

Cheaper lights use a resistor to control the current through the diode, but this wastes energy. The pulsing circuit increases battery life.

Brightness of LED sources is controlled with PWM (Pulse Width Modulation) -- the LED flashes (time_on/time_of - duty cycle). Due this 'effect' you see rain drop multiple times with gaps. Human eye is capable distinguish movie 'frames' with frequency below 24Hz (nowadays a good TV provides 120Hz refresh rate).