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No. Premium processing is not available for H-1B1, and it never has been. Premium processing is the service under which USCIS, for an extra fee, guarantees that it will take adjudicative action on a case within a set period, 15 business days for the nonimmigrant classifications filed on Form I-129, or refund the fee. It is requested on Form I-907, and it is available only for the benefit requests that the regulations designate.
The exclusion is a matter of how the designation is written. The list of eligible benefit requests appears at 8 CFR 106.4(c), and its specialty occupation entry reaches only a nonimmigrant described in INA 101(a)(15)(H)(i)(b). H-1B1 professionals are described in subclause (b1) of that same provision, a separate classification, and no other paragraph of the list picks them up. The table on the USCIS Form I-907 page matches the regulation: it lists E-1, E-2, E-3, H-1B, H-2B, H-3, L-1A, L-1B, blanket L, O-1, O-2, the P classifications, Q-1, R-1, TN-1 and TN-2, with no H-1B1 line.
The gap is peculiar to H-1B1. The comparable agreement-based specialty categories do qualify. TN professionals from Canada and Mexico have been eligible since 2001, and E-3 professionals from Australia were added on February 24, 2021. H-1B1 is the one specialty occupation classification of that group left out.
In practice the point arises less often than it might appear. H-1B1 is not petition-based, so the initial visa is applied for at a U.S. consular post and USCIS is not involved at all. The question matters only for a petition filed inside the United States to extend H-1B1 status or to change into it, and those filings must take regular processing times. Where speed is essential, the remaining routes are a USCIS expedite request under the agency's published criteria, or applying for the visa at a consular post abroad. An employer filing a genuine H-1B petition for the same worker may use premium processing for that petition, because the H-1B classification is designated.
The designated list does change over time, as the addition of E-3 shows, and the fee is adjusted for inflation on a biennial basis. Anyone relying on this answer should confirm the current Form I-129 table on the USCIS premium processing page before assuming the position still holds.