Disable SCK temporarily to save power
Re-visiting the issue, I have found a simple answer:

I set the SPI pins as outputs and inputs respectively:

const int chipSelect = 10;
const int MOSIPin = 11;
const int MISOPin = 12;
const int SCKPin = 13;


and

  pinMode(chipSelect, OUTPUT);
  pinMode(MOSIPin, OUTPUT);
  pinMode(MISOPin, INPUT);
  pinMode(SCKPin, OUTPUT);


Now, before I power down my machine, I do the following:

  SPI.end();
  digitalWrite(SCKPin, HIGH);
  digitalWrite(chipSelect, HIGH);
  digitalWrite(MOSIPin, HIGH);


Now, the drawn current drops from 1mA to roughly 0,02mA, which is satisfactoy for me!

Thanks for reading!