Yes - if you're registered for PAYE and paying yourself a salary each month, you file an FPS on or before each payday, even with just one director-employee. GOV.UK's rule is that the full payment submission must be sent on or before the payday, and being the sole director doesn't change that. 'Small' isn't a category in the rules: if the salary is on the payroll, it's reported monthly like any other payday.
Directors' NIC is worth handling properly here. Directors can be assessed on an annual or cumulative basis rather than per pay period, which avoids small monthly NIC blips on a director's salary - but that's a NIC calculation choice, not an exemption from filing. The FPS still goes every month.
Could you switch to an annual payroll scheme instead? If you pay yourself just once a year (a single annual salary payment), you can register as an annual PAYE scheme with HMRC and file one FPS a year. That's a legitimate option for director-only companies - but it only works if you genuinely pay yourself once. With a monthly salary as now, it's a monthly FPS.
Penalties are real and worth knowing: for employers with 1 to 9 employees, the late-filing penalty is £100 per month, so a director-only payroll faces the same £100 monthly penalty as a bigger employer. There is some grace - HMRC doesn't penalise if the FPS arrives within 3 days of the payday, and the first failure in a tax year isn't penalised - but relying on that is poor planning. And if you stop filing altogether, HMRC can issue 'specified charges' based on what it thinks you owe, which you'll then have to sort out.
You also can't just run it through Self Assessment instead: once a PAYE scheme exists and salary is paid, RTI reporting is the mechanism, and Self Assessment covers your personal tax on the salary, not the payroll filing.
Practically: keep monthly FPS filings, issue yourself a payslip on or before each payday, and keep the records for three years from the end of the tax year. One caveat - if your pay is below the lower earnings limit and you have no other PAYE triggers, you might not need a scheme at all; but once registered, full RTI applies.