So far all the victims, aged from 16 to 39, have escaped with minor injuries, although at least one required stitches.
Police fear that if the attacks continue the man will seriously harm or kill someone. The man plunges a screwdriver-type weapon into his victims' backs as he speeds past on his bicycle.
The latest assault - the fifth within a week - happened on Wednesday afternoon as a 22- year-old student was walking in the town centre.
The victim, who did not want to be named, said last night: 'Suddenly I felt this severe blow on my back and this cyclist shot past me. My body jolted, I lost my breath and this pain shot up my back and down my front.'
It was only when she got home she realised she had been stabbed and found a small wound between her shoulder blades. 'I think I was lucky with my injuries and I just want him caught before he really hurts some poor woman,' she said.
The cyclist, who has used a mountain bike and a racing bicycle during his assaults, first struck in January. His victims appear to be chosen at random. In one incident a 39-year-old woman was stabbed while cycling.
Since Wednesday police officers have carried out special patrols in Bedford to search for the attacker.
Detective Inspector David Tomlinson, who is in charge of the investigation, said the assailant seemed to be getting more confident.
'These attacks are unprovoked and highly dangerous,' he said. 'If this man hits one of these women in the neck the consequences could be terrible.
'Women in Bedford should not be frightened of going out, but everyone in the town should be aware of the possible danger. Someone must have information about him, or have seen something suspicious.'
Police described the attacker as aged 18 to 20, tall, with broad shoulders and blond, collar length curly hair. He usually wears a hat or hood.