A number of Church of England bishops have broken ranks to distance themselves from guidance issued last week which said only married heterosexual people may have sex and those in civil partnerships should be celibate.

Rachel Treweek, the bishop of Gloucester, issued a statement saying she was “deeply frustrated and saddened” at the guidance.

She said: “I recognise that it has fanned into flame unnecessary pain and distress and I wish to acknowledge my part in that.”

In terms of factual content, the guidance – which she saw last month – reiterated the C of E’s doctrinal position, she said. But its publication “in cold isolation from anything else … has been perplexing and upsetting”.

She continued: “The word ‘love’ emanating from the generous love of God is one that needs to be heard and lived, and I am extremely sorry that it has not been heard in the publication of the House of Bishops’ statement.”

Pete Wilcox, the bishop of Sheffield, described Treweek’s statement as “very helpful” on Twitter. John Inge, the bishop of Worcester, tweeted: “I echo all that Bishop Rachel says.” David Walker, the bishop of Manchester, and Rob Wickham, the bishop of Edmonton, also retweeted Treweek’s statement.

The bishops of Bristol, Newcastle, Norwich, Repton, Crediton and Tewkesbury have also supported Treweek’s statement. Olivia Graham, the bishop of Reading, tweeted that she was “deeply saddened by the unpastoral tone” of the guidance, saying it was cold, legalistic and loveless and that “bishops are not of a mind on this”.

Meanwhile, Paul Bayes, the bishop of Liverpool, retweeted an open letter saying the church’s stance had made it a “laughing stock”, suggesting “here is one way to comment” on the bishops’ guidance.

The open letter had been signed by more than 3,000 people by Monday evening, including more than 800 clergy, one serving bishop, eight retired bishops and 10 serving deans. It has been sent to the archbishops of Canterbury and York, plus Stephen Cottrell, the bishop of Chelmsford, who will take over as archbishop of York in the summer.

The signatories express “anger and disappointment” at the bishops’ statement, saying: “It is cold, defensive, and uncaring of its impact on the millions of people it affects.”

The bishops’ guidance was issued in response to the recent introduction of mixed-sex civil partnerships. It said: “Sexual relationships outside heterosexual marriage are regarded as falling short of God’s purpose for human beings.”

The church sought “to affirm the value of committed, sexually abstinent friendships” within civil partnerships, it added.

It was published ahead of the conclusions of a major review of the C of E’s stance on sexuality and marriage, which is due to be delivered this year.

The church is deeply divided on the issue, with progressives arguing for inclusivity and equality and conservatives stressing traditional biblical teaching on sexuality and marriage.