Rona Fairhead should resign or be sacked as chairman of the BBC Trust, a leading MP demanded yesterday.

Margaret Hodge said Mrs Fairhead, who is also a director of HSBC, had been either ‘incredibly naive or totally incompetent’ over the tax evasion scandal at the bank’s Swiss division.

The 53-year-old was criticised by MPs on the powerful Public Accounts Committee as they grilled her over allegations that the embattled bank helped wealthy clients to hide billions of pounds from the taxman.

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Rona Fairhead faced a call to resign as chairman of the BBC Trust after being grilled about her time as a non-executive director on HSBC's board

Mrs Fairhead claimed repeatedly that she had not been aware of any wrongdoing and that she had relied on various committees and structures to flag up problems.

But MPs said her performance at the bank was not good enough – raising serious questions over whether she is suitable for her BBC role.

Committee chairman Margaret Hodge said: ‘I really do think you should consider your position and consider resigning and if not, the Government should sack you. Either you knew or you didn’t know – and I think in that case that you are either incredibly naive or totally incompetent.

‘I don’t think that the performance you’ve shown here, and your record as a guardian of HSBC give me confidence that you should be the guardian of the licence fee.’

Labour MP Mrs Hodge added: I’ve met you once and I like you but I don’t think you are fit for the job… I’m afraid you’ve lost my trust.’

Mrs Fairhead has already come under considerable pressure over potential conflicts between her multiple jobs, since she started as chairman of the BBC Trust last September.

Committee chairman Margaret Hodge said Ms Fairhead had lost her trust, but Tory committee member Stephen Hammond disagreed with the call for her to resign

In addition to the £110,000-a-year position, on which she spends three days a week, she is a non-executive director of Pepsi and HSBC.

Last year she received more than £513,000 in fees and benefits for her work at the troubled bank, including a £334,000 fee as non-executive chairman of HSBC North America Holdings.

Yesterday, MPs marvelled at the fact that she is paid so much, but did not do more to find out about HSBC’s alleged efforts to help tax evaders.

Mrs Fairhead was one of a panel of HSBC bosses hauled in front of the committee who spent the session trying to blame others for the tax evasion scandal at its division in Switzerland.

Leaked files showed that more than 1,000 people from the UK had secret accounts at HSBC’s Swiss bank, including a number of Tory donors.

The Public Accounts Committee of MPs was taking on Tax avoidance and evasion: HSBC in Parliament

But she argued that the chief culprits were those who had evaded tax and people on the front line, rather than the non-executive directors.

‘We relied on three layers of board oversight and multiple layers of executive oversight to raise issues of material importance to the audit committee,’ Mrs Fairhead said. ‘We were reliant on [reports from] specialists… It wasn’t flagged so you’ve got to ask, “What more can you do?’’’ MPs responded that the BBC Trust chief sounded too ‘passive’, and accused the wider HSBC board of ‘extraordinary’ blindness.

Last week Mrs Fairhead told a conference that she would prioritise her job at the BBC Trust over her other positions, and that she was keeping her roles on the HSBC and Pepsi boards ‘under review’.