Lithium Iron Phosphate --LiFePo4 --batteries don't off-gas.
Unless you are serious about refusing to allow any lithium products on your boat (including cordless tools, handheld VHFs, iPhones, laptops --all of which have the MUCH more unstable Li ion chemistries that actually can dramatically catch fire), I think you should do your own research, and not rely on myths.
Safer lithium products--like LiFePRO4--have been designed for safe use on boats and are pretty darned safe.
In my opinion based on my research based on actual data -- not hearsay and WAGs, this specific lithium chemistry (LiFePO4) is designed for stability and safety. The products come in various capacities and are designed to be suitable for the marine environment. They are sealed and consist of waterproof cells inside of waterproof cases. The added benefits of Lithium, other than more KWHs, longer lifespan, and stability over lead specifically for a flybridge battery backup application are that LiFePO4s can hold a charge a long time (they self-discharge very slowly) so they don't need a trickle charger--just a periodic (annual) voltage check and possible top up. Their voltage profile when discharging is much flatter than lead acid so you can run everything at nominal voltage until they are out of juice. I have added a switched backup battery (50AHr, 12v) under my helm for emergency use in the event of total electrical system failure. It will power one VHF, my AIS, one garmin display, and provide USB and recharging ports for my phones, tablets and laptop for ~8 hours emergency use. Much less risk than using those Li Ion power supplies that some people use that sometimes try to cook themselves.
Two cautions with LiFePO4--Cold weather ops and wiring/circuit protection:
You should never charge them when in freezing weather, when the battery temp is <32degF/0degC. (You can use/discharge them fine at low temps, just not recharge them without damaging them). For up in a flybridge, make sure you purchase batteries that have low temp charging protection if you plan to trickle charge. Also, you'll need to charge them on a charger designed for Lithium charge profile.
Additionally, lithium batteries are very low resistance which means they must be protected with appropriate fuses. The vast majority of safety issues involving upgrading to Lithium are overheats and overloads not of the lithium battery, but overheating/frying boat wires because of poor fusing or loose terminals. You can't just replace a lead acid battery with an equivalent sized "drop-in" LiFePO4 and think you're all set without understanding how the lithium battery affects other electrical components (chargers & alternators in particular).
But adding a switched lithium battery backup to helm electronics is an easy application for lithium and makes a lot more sense than lead acid options.