The mother was spotted ignoring the danger signs beside Rushmere lake on Wimbledon Common, south-west London.
Onlookers said she and the girl spent about ten minutes on the frozen water, on their way to a ?3,000-a-term preparatory school, The Study.
Two teenage boys were later seen leaping up and down on the same ice-covered lake, spending about 45 minutes trying to crack the surface ? before eventually giving up.
Another used a large wooden stick in similarly fruitless efforts to break through.