Many reported the heavy rain and buffeting winds on the Spanish island was a tornado.
The Met Office could not say for certain that it was, but added that atmospheric conditions were conducive for one to form.
Sophie Kelly, who is on holiday in Ibiza with her son Riley, said the storm struck quickly, "snapping trees in two" and causing heavy flooding.
"The storm was brewing for about an hour, then it went very dark and the rain started, then the wind," she said.
"The full force of it only lasted about four minutes maximum.
"That's why I assumed it was a tornado, plus the way the trees are snapped in a narrow strip."
Ms Kelly, from Newquay in Cornwall, said lightning continued during last night and was accompanied by power cuts throughout the evening.
She said many people were now leaving the hotel she is staying at, where staff used chainsaws to cut up fallen trees this morning.
Peter Nixon, 56, lives in Altea on the Spanish mainland to the west of Ibiza and watched the storm rolling over the area from afar.
"The storms are known locally as the Goto Fria and signify a change in the weather from summer to winter," he said.
"They always seem to hit on UK half-term week, which is a bummer for the many holidaymakers with kids."
The Met Office said the storms are expected to continue this week, moving east across the Mediterranean and bringing very heavy showers, hail and thunderstorms, with up to 8 inches (200mm) of rain expected in some areas.