Southeastern is unlikely to be able to run any metro or mainline services until at least midday tomorrow, the rail operator has said.
"Network Rail will be out at first light checking the track for any damage, flooding and fallen trees and will advise us when it is safe to start running train services," a statement on Southeastern's website said.
Southeastern said it expects to be able to run some high-speed shuttle services between Ashford and St Pancras International in the morning, but advised passengers to check before setting out.
Southeastern reports that services remain disrupted across its entire network due to the poor weather:
It emerged that Miss Banks had earlier been seen at Tunbridge Wells Hospital but was sent away as she was not in labour and contractions had not started.
Commuter Neil Perkins, 22, a recruitment consultant from Maidstone, was sitting close to Miss Banks before she went into labour. He said his "jaw dropped" when he realised she had given birth.
The father of a baby girl born on a train packed full of commuters during rush hour in Kent has said: "It feels like I'm on cloud nine."
Allan Stanley, 21, was travelling back to his mother's home with her and his partner Sonia Banks, 22, when the contractions started.
Moments later, on Thursday night's 18:18 London to Kent train, Miss Banks gave birth to a "perfectly healthy" 6lb 13oz baby called Phoebe.
A man claiming to be the uncle of baby Phoebe, who was born on a commuter train from London Victoria to Kent yesterday evening, has tweeted to say his niece is "well and safe."
A "serious fire in an electrical substation" is causing major disruption on the Southeastern train network, the company said on their website.