No – If your PA is below the earnings threshold for SSP, then this would be unpaid leave, consider and/or discuss Annual Leave or benefits. More links below.
Your Personal Assistant (PA) will only be eligible to received SSP if they earn over £120 per week. If your PA hours vary – work out an average of their pay over the last 12 weeks – if this is £120 or over, they are eligible for SSP. If it is less than £120 per week, they are not eligible for SSP and it would therefore be UNPAID leave.
You can keep up to date on Gov.uk SSP link: https://www.gov.uk/statutory-sick-pay
Benefits that maybe available:
Employment Support Allowance (ESA) https://www.gov.uk/employment-support-allowance
Universal Credit https://www.gov.uk/universal-credit