Yes and No. They are technically your employer but due to low earnings and a personal allowance they don’t need to pay tax if your allowances offset any taxable amount.
However apart from income tax there is also national insurance, and you are credited with national insurance contributions to ensure continuity of contribution while you are out of work and hence retain certain pension benefits when you reach retirement, as they based on total years of contribution.
You can always ask HMRC for a statement of national insurance contributions. This will show what years have full contributions, which have part and which have none. This is useful as there is about a 5 or 6 year period to catch up on any shortfall by paying voluntary contributions towards your pension entitlement. Individual’s circumstances may vary and it may depend on how close to retirement you are, for some it makes sense paying any shortfall but for others it does not.