Yes, provided the citizen spouse owns at least half of the enterprise and is the principal applicant, so that the business takes its treaty nationality from a Canadian citizen. The permanent-resident spouse can then be admitted as an E-2 dependant and is employment-authorized.
Nationality follows the owner, so structure the ownership
E-2 treaty nationality attaches to the individual investor and to the enterprise, and the enterprise's nationality is determined by the nationality of the people who own at least fifty percent of it. A Canadian permanent resident has no treaty nationality through Canada; his or her own citizenship, if from another treaty country, might supply it, but the simpler route in this case is to place the controlling interest with the citizen spouse. A fifty-one-forty-nine split, or sole ownership by the citizen, both work; a fifty-fifty split can also work if the citizen spouse has equal management rights that provide negative control, but those rights should be documented before filing. The permanent-resident spouse is admitted as the E-2 spouse and is authorized to work incident to status, so both can bake, sell and manage. Children may study but not work, and derivative status ends at twenty-one, which matters more here than in TN cases because E-2 can be renewed indefinitely while the business operates.
Hypothetical example: a citizen spouse owns sixty percent of a bakery and the Canadian permanent-resident spouse owns forty percent and plans to supervise morning production. The ownership register, passport and operating agreement settle treaty nationality and control; a permanent-resident card does not substitute for the required nationality. The family review should then list each derivative's relationship, age, passport and intended activity. A qualifying spouse may have work permission connected to the classification, but a child cannot take bakery shifts merely because the business is family-run. A common error is treating a fifty-fifty ownership split as automatically clear without examining which nationalities are represented and who will direct the enterprise. Resolve the ownership and management documents before submitting the application, because a later reallocation may change the facts that supported eligibility.