There are several ways to handle this. Without seeing your code, I can only guess at what you've tried.

One approach is to read more than a line at a time. You could set $/ to read in a record at a time, which is made easier if there's a blank line or some other marker between records.

Another approach is to read the entire file into a scalar, by localizing $/ within a block, reading the file into a scalar, and then using multi-line regexes.

You could read the entire file into an array, loop through with an index variable, incrementing the index as necessary, or building up a second array of lines.

Similarly, you could loop through a line at a time, building a single array, appending a continued line.

Depending on how large the file is, you might want to continue processing a line at a time. I would probably do something like this pseudo-code:

# open file
# declare a last_line variable
# loop, reading a line at a time
# check to see if line is continued
# if so, append the current item to the last_line variable
# set a flag that we've done this
# process last_line (in another function, probably)
# stick the current line in last_line
# if the flag is set, empty out last_line
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This depends on a lot of specifics, but you get to process (basically) a line at a time for the small overhead of keeping a little bit of state information around.