No. L derivatives are not subject to a nationality test, unlike TN and E-2, so a spouse who is a Canadian permanent resident and a citizen of another country is unaffected. The L-2 spouse is employment-authorized incident to that status, and unmarried children qualify while under 21.
L is the category that does not ask where you are from
Hypothetical example: Valerie is transferring on L-1B. Her husband Denis is a Canadian permanent resident and a citizen of Algeria, and their son is seventeen. In a TN or a treaty-investor file Denis's citizenship would matter a great deal, because both of those categories test nationality directly.
L does not. The derivative spouse and children of an L-1 principal are admitted on the relationship, and Denis's Algerian citizenship neither helps nor hinders. He should still hold his own valid travel document and be prepared for the ordinary consular process that applies to citizens of his country.
Two further points are worth writing down. Permanent residence is not citizenship, and Denis should not describe himself as Canadian on any form, because the correction is far more awkward later than the accuracy is now. And the son's derivative status ends when he turns 21, which is four years away and should be planned for rather than remembered.
- USCIS: L-1B intracompany transferee specialized knowledge
- USCIS Policy Manual, Volume 2 Part L: intracompany transferees
- USCIS: L classification documentation and evidence
- 8 U.S.C. 1184: admission of nonimmigrants and the specialized knowledge definition
- Department of State: 9 FAM 402.12, intracompany transferees
- USCIS: Form G-1055, fee schedule