Yes, trekkers need a special permit from the Kyrgyzstan government to hike to Lenin Peak Base Camp and it costs around $15 for the regular application time (1-3 weeks) and $30-$50 for expedited services (e.g., 1-3 days). It is important to obtain the permit before embarking on the trek to Base Camp One. The permit can be obtained through a local tour company.
Note: Visit Alay includes the application process and cost of the permit (for regular speed applications) for Lenin Peak Base Camp as part of the tour price.