Death threats were made against Gibraltar's chief minister Fabian Picardo following his criticism of Spain's imposition of heightened border checks, which have caused delays of up to five hours to enter or leave the territory.
Conservative MEP Julie Girling, whose South West England constituency includes Gibraltar, said she had been called a Nazi along with other insults, in a series of obscene messages including one which said she was "too ugly to visit Spain anyway".
"I have had some extreme reactions to things I say. I have even had death threats against my family by Twitter," said Picardo, who is married to the lawyer Justine Olivero, with whom he has a young son.
Asked where the messages came from, Picardo said: "From Spain, in Spanish."
He added: "Social media is a great thing, but it also has a very negative aspect to it, which is people can use the cloak of anonymity to insult people and to say things that reasonable human beings don't say about each other."
Girling said she had received "a series of deeply offensive and racist messages" on Twitter.
She had called on British holidaymakers to boycott Spain over the Gibraltar row.
Girling said one user, calling himself Javier, wrote: "You are a Nazi Julie Goering."
Another used racist language to suggest she should holiday in India.
Girling said: "It may be that to be trolled goes with the job. However, the Spanish Twitter trolls aren't doing their argument any favours by trolling me.
"Their abuse shows they have no argument as they can only resort to the lowest form of language. The people of Gibraltar need our support.
"They will not be bullied by the Spanish Government or by a minority of the Spanish people. Neither will I."
Picardo has blamed Gibraltar's problems on Madrid, saying locals Spaniards were supportive of Gibraltar as an "economic engine" and wanted border restrictions eased.
He warned that a protracted blockade could harm Spain's struggling economy, with construction jobs at particular risk.
Picardo said: "I have no contact with Madrid, but I have a lot of fluid contact with mayors in La Linea and others municipalities in the area who are very concerned about the effect the controls being imposed by Madrid could have on the working lives of people who come in and out of Gibraltar every day.
"So often, the people of Gibraltar and the people of La Linea and San Roque become the whipping boys for what may be happening in Madrid, which might be totally unrelated to us here.
"The municipalities in the area are hugely supportive of Gibraltar continuing to be an economic engine in the zone and the free flow of people and goods to and from Gibraltar is an important aspect of that.
"For example, we expect to be putting up a lot of affordable housing in the next two years, we would bring in a lot of Spanish labour for the purposes of building that housing.
"Of course if they can't get across the frontier we will have to bring in labour from elsewhere."
The Foreign Office has lodged a formal complaint with Spain over the delays at the border.
The move to increase border security by Spain came in response to Gibraltar's construction of an artificial reef, which Madrid says has harmed fishing in the area.
In separate incidents, three cars bearing Gibraltar registration plates were set alight on the Spanish mainland amid local tension over the row.
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