What you're describing is usually called a core dump. Your kernel has the ability to trigger these built-in, and it's a common thing to be done automatically (through systemd-coredump) when your program segfaults.
You can also make a core dump manually, using a debugger. gdb is pretty much the standard debugger under linux:
shell> gdb --args /path/to/your/executable --sensible-arg1 -s 2 --sensible-arg3 foobar
[…]
(gdb) start
Temporary breakpoint 1, main(argc=6, argv=…)
(gdb) break the_function_at_which_you_want_to_break
(gdb) continue
[…]
(gdb) generate-core-file
Saved corefile core.123456
(gdb) quit
Now you have a corefile in which you can operate to your liking.
Honestly, I find radare2 to be the tool of choice to inspect process memory (you can do the same as above directly in radare2)
shell> radare2 -d /path/to/your/executable --sensible-arg1 -s 2 --sensible-arg3 foobar
[0x7f44444]> aaa
[x] Analyze all flags starting with sym. and entry0 (aa)
[x] Analyze function calls (aac)
[x] Analyze len bytes of instructions for references (aar)
[x] Finding and parsing C++ vtables (avrr)
[x] Skipping type matching analysis in debugger mode (aaft)
[x] Propagate noreturn information (aanr)
[x] Use -AA or aaaa to perform additional experimental analysis.
[0x7f44444]> db sym.the_function_at_which_you_want_to_break
[0x7f44444]> dc
[0x7f44444]> e search.in=dbg.maps
[0x7f44444]> / the_needle_you're_looking_for
0x7f524662c3ed hit1_0 some_more_dirty_stuff the_needle_you're_looking_for
…
[0x7f44444]> s hit1_0
[0x7f524662c3ed]> pr
some_more_dirty_stuff the_needle_you're_looking_for Things go really downhill from here…
Radare2 is nice because it's really meant for memory analysis, and looking for things like cryptographic secrets in memory. For example, it has an excellent entropy display, with which you can often spot things like secret keys in binaries, or compressed data in memory.
On the flip side, I had to look up all the commands myself... it's a highly specialized tool with it's own mechanics.
