The display driver is not 5V tolerant. Put series resistors in the control and data lines as follows:
pro mini digital pin 7 through a 1K2 resistor to display RST
pro mini digital pin 8 through a 1K2 resistor to display DC
pro mini digital pin 9 through a 1K2 resistor to display CS
pro mini digital pin 11 through a 1K2 resistor to display MOSI
pro mini digital pin 13 through a 1K2 resistor to display SCLK