The French were in the Americas before the British--and by British, as you pointed out, we should really be talking about the English or the Scottish here since "Great Britain" wasn't formed until the Acts of Union of 1707.

Even then, the French were settling the Americas before the English or the Scottish (or the Welsh or the Irish for that matter).

Both the French and the English had some failed colonies before they established one that was successful. The English famously failed with the Roanoke colony, founded in 1587 but had been abandoned by 1590.

Earlier than that, a site near present day Quebec City was first settled by the French as a seasonal colony in 1535, though it was abandoned by the 1540s.

France's first year-round colony in the Americas was France Antarctique, established near present day Rio de Janeiro in 1555 under the leadership of Nicolas Durand de Villegaignon. This colony lasted until 1567, when it was conquered by the Portuguese.

In North America, the French founded Fort Caroline in 1564 near present day Jacksonville, Florida, but it was attacked by the Spanish in 1565 and destroyed.

As for settlements that survived, yes, the oldest English settlement is Jamestown, Virginia, dating to 1607. However, the French have had year-round residents in North America longer. Samuel de Champlain first established a colony called St.-Croix in 1604 in present day Maine, but the winter was harsh and they relocated to present day Nova Scotia in 1605, founding a colony called Port-Royal, Acadia.

Champlain returned to France, leaving permanent colonists there at Port-Royal, and returned to the New World in 1608 when he founded the city of Quebec. Due to some religious/political disagreements at Port-Royal, several of the Port-Royal settlers left and joined Champlain's new recruits in Quebec.

Both the settlements of Port-Royal and Quebec co-existed until 1613. That year, a few missionaries from Port-Royal left for Mount Desert Island off the coast of Maine to establish a mission to convert local Indians. But within a few months, the English attacked this new settlement, killed the missionaries, destroyed their buildings, sailed up to St.-Croix and destroyed the remaining French buildings there, and then sailed up to Port Royal and destroyed the fort there, too. The French settlers of Port-Royal were either killed or taken prisoner. The survivors were eventually released and made it back to France.

In 1615, Port-Royal was re-established a few miles/kilometers downriver from the original by Charles de Biencourt de Saint-Just and his father Jean. That site still exists and has remained continuously populated by Europeans ever since, with its name changed to Annapolis Royal along the way.

But if you follow the trail of Samuel de Champlain, settlements founded under him have had continuous French residents in North America since 1604, although it wasn't until 1608 and the founding of Quebec that those settlements have remained in the same place.

Sources:

Pioneers of France in the New World by Francis Parkman

Builders of Canada from Cartier to Laurier ed. by Thomas Guthrie Marquis

A Popular History of the Dominion of Canada from the Discovery of America to the Present Time by William Henry Withrow

Voyages of Samuel de Champlain 1604-1618 by Samuel de Champlain

The History of Canada: From Its First Discovery to the Present Time, Vol. 1 by John Mercier McMullen

Further reading:

Champlain's Dream by David Hackett Fischer

Champlain: The Birth of French America by Raymonde Litalien