Is there a way to get the user's SSH client IP address in a bash script?

From the manual of OpenSSH SSH client (man 1 ssh):

ENVIRONMENT
ssh will normally set the following environment variables:

[…]

SSH_CONNECTION Identifies the client and server ends of the connection. The variable contains four space-separated values: client IP address, client port number, server IP address, and server port number.

[…]

In .bashrc you should do like:

case "${SSH_CONNECTION%% *}" in
1.2.3.4|1.2.3.5 )
   PS1=…
   ;;
2.2.7.* )
   PS1=…
   ;;
# add support for more users if needed
esac

or like

if [ "${SSH_CONNECTION%% *}" = "1.2.3.4" ]; then
   PS1=…
else
   PS1=…
fi