Babbler: It's not very likely outside of young children and the very old... maybe in the hinterland of the Balearics or up in the Pyrenees*? It's far more common, I think, that there are Catalans who only speak rather bad and imperfect Castillian.
The thing to remember with Basque is that it only has about 30% or 40% penetration in Spanish Euskadi, even the father of Basque nationalism, Sabino Arana, had to learn the language in adulthood, and within early Basque nationalist circles there was a serious debate about whether to abandon the language in favour of Castillian or revive Basque in the urban communities where it had been displaced. A Basque-speaking polity has always been more dream than reality, and although it's doing better these days, it's not as successful as the Catalan project.
I would be willing to bet, however, that there are still monolingual Galicians in some villages in Galicia, because Galician is more universally spoken than Basque or Catalan in their respective homelands and the place is, frankly, far less modern and developed. Gallego has something like 90% spread in Galicia's Autonomous Community, versus something like 60% spoken fluency in Catalonia and 30% Basque in the Basque Country.
*There's even people whose first language is Aragonese in their part of the Pyrenees. Shame their language didn't become the peninsular standard - its orthography is even simpler than Castellano.