The MP for East Dunbartonshire was found collapsed at home by her partner, and has undergone emergency neurosurgery.
The statement said: “Thanks to the immense skill and response of the medical team involved in her care, Amy was able to successfully pull through the operation and is now stable in hospital.”
Thanking NHS staff for “saving her life”, it added: “Amy is in good spirits and able to communicate well with family. She has been overwhelmed with the messages of support and kindness, which are a huge source of encouragement.
“She has made great progress thus far but is under no illusion of the seriousness of her condition and the rehabilitation that will follow.
“As she always has done throughout her life, she will continue to fight. She is determined to overcome what barriers, if any, may now arise as a result of this incident.
“She wants to make clear that her constituency work will continue: her office is still open for all those who require help and support.
“There will be a period of required rest and recovery. Whilst this is ongoing, we ask that you respect her and her family’s safety and privacy.
“It is the privilege of her life to be elected as the Member of Parliament for East Dunbartonshire.
“There is absolutely no doubt that Amy will come back stronger, fitter and more determined than ever to continue in that role and serve, to the best of her ability, the people of her constituency.”
SNP Westminster leader Ian Blackford said: “My thoughts and best wishes are with Amy and her family at this incredibly difficult time.
“I wish Amy a full and speedy recovery from everyone in the SNP, and would ask that people please respect her family’s privacy now and in the weeks to come.”
First Minister Nicola Sturgeon said she was sending “lots of love and strength” to Ms Callaghan and her family.
“Wishing her the speediest possible recovery,” she posted on Twitter.
Ms Callaghan was elected in the 2019 general election, unseating former Liberal Democrat leader Jo Swinson by just 149 votes.